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LIBRARY  OF 

ALLEN  KNIGHT 

CERTIFIED  PUBLIC  ACCOUNTANT 

502  California  Street 

SAN     FRANCISCO.     CALIFORNIA 


GIFT   OF 


OsilSliw- 


N^WvMD^t^ 


LOGARITHMS 

TO  12  PLACES 

AND     THEIR      USE      IN 

INTEREST  CALCULATIONS 


By  CHARLES  E.  SPRAGUE 
Author  of  "Thb  Phii^osophy  of  Accounts" 
'•Text-Book  of  the  Accountancy  of  Investment 
and  "Extended  Bond  Tabi^es" 


New  York,  1910 
publisht  by  the  author 


s^ 


^-7 


Copyright,  1910,  by  Charles  E.  Sprague. 


^  Oid^  \t^^sA3r 


TRUNK  BROS. 

18    FRANKFORT    ST. 

NEW  YORK 


PREFACE. 


The  need  of  a  logarithmic  table  for  special  cases,  where 
the  usual  five-figure  and  seven-place  results  are  insufiBcient,  is 
often  felt  by  the  accountant  and  the  actuary.  Rough  results 
will  answer  for  approximativ  purposes  ;  but  where  it  is  desir- 
able, for  instance,  to  construct  a  table  of  amortization,  sinking 
fund  or  valuation  of  a  lease  at  an  unusual  rate,  for  a  large 
amount  and  for  a  great  many  years,  exactness  is  desirable  and 
becomes  self-proving  at  the  end. 

It  is,  of  course,  a  slower  process  than  that  for  a  few  places, 
but  as  the  figures  from  which  all  results  are  obtainable  are 
containd  in  two  pages  instead  of  200,  there  is,  on  the  other 
hand,  a  great  saving  in  the  mechanical  labor  of  turning  leaves. 

It  also  contains  a  thoro  analysis  of  the  entire  doctrin  of 
interest,  explaining  every  process  by  the  use  of  logarithms,  as 
well  as  arithmetically  and  algebraically. 

CHARLES  E.  SPRAGUE. 


Union  Dime  Savings  Bank, 
New  York,  January,  1910. 


380313 


TABLE  OF  CONTENTS. 


PART  I. — Thk  Properties  of  Logarithms. 

Page 

The  Nature  of  Logarithms 1 

Multiplication  by  IvOgarithms 3 

Division  of  Logarithms 5 

Tables  of  Logarithms 6 

To  Find  the  Number 8 

To  Form  the  Logarithm 16 

Less  than  12  Places 21 

Multiplying  Up 22 

Signs  of  the  Characteristics 27 

Different  Bases 28 


PART  II. — Tables  for  Obtaining  Logarithms  and 
Antilogarithms  to  12  Places  of  Decimals. 

Table  of  Factors 30 

Table  of  Interest  Ratios 32 

Table  of  Sub-Reciprocals 33 

Table  of  Multiples 34 

Logarithmic  Paper 36 


PART  III. — The  Doctrin  of  Interest. 

Definitions 39 

The  Amount 41 

The  Present  Worth 43 

The  Compound  Interest  and  Discount 45 

Finding  Time  or  Rate 46 

The  Annuity 47 

Amount  of  Annuity 48 

Present  Worth  of  Annuity 50 

Amortization 52 

Special  Forms  of  Annuity 53 

The  Unit  of  Time 55 

Frequency  of  Payment 57 

Coefficients  of  Frequency 58 

Fractional  Periods 63 

Sinking  Funds 65 

Interest-Bearing  Securities 67 

Multiplying  Down 72 

Computing  Amortizations 73 

Discounting 75 

Intermediate  Purchases 76 

Intermediate  Balances > 77 

Short  Periods 80 

Finding  the  Income  Rate ...    83 

Interest  Formulas 86 


PART    I 


The  Properties  of  Logarithms 


PART     I. 
THE  PROPERTIES  OF  LOGARITHMS. 


1.  — If  we  multiply  5  lO's  together,  10  x  10  x  10  x  10  x  10, 
we  may  write  the  result  as 

100000 
or     10^ 

or     the  fifth  power  of  ten. 
The  little  "  ^  "  is  the  exponent  of  the  power.     We  may 
form  a  series  of  the  powers  of  10  : 


00000   or   10^ 

10000 

10* 

1000 

10« 

100 

10« 

10 

10^ 

1 

10° 

2. — The  following  observations  may  then  be  made : 

1.  The  number  of  the  zeroes  in  the  first  colum  is 
the  exponent  in  the  second. 

2.  Each  term  in  the  first  colum  is  one-tenth  of  the 
one  above  it,  while  in  the  second  colum  each  exponent  is 
one  less  than  the  exponent  above  it.  This  leads  to  the 
result  that  10°  =  1,  which  at  first  seems  paradoxical. 

3.  If  we  multiply  together  any  two  terms  in  the  first 
colum,  we  add  the  exponents  in  the  second. 

3. — Logarithms  are  auxiliary  numbers  having  relation 
to  a  base.  When  the  base  is  once  fixt,  every  possible  number 
has  its  logarithm.  The  customary  and  most  convenient  base 
is  10,  because  our  whole  system  of  numeration  is  based  upon 
ten.  The  logarithms  are  simply  exponents  and  we  re-write 
the  above  series  thus  : 


Thk  Properties  of  Logarithms. 


The  base  being  10, 

100000  is  the  number  whose  logarithm  is  5 

or  contracted, 


100000. 

nl 

5 

10000. 

nl 

4 

1000. 

nl 

3 

100. 

nl 

2 

10. 

nl 

1 

1. 

nl 

0 

.1 

nl 

— 1 

.01 

7ll 

-2 

.001 

nl 

-3 

.0001 

nl 

-4 

.00001 

nl 

-5 

The  copula  {nl)  means  "is  the  number  whose  logarithm  is ;" 

while  {In)  means  "is  the  logarithm  of  the  number ." 

4. — We  have  here  logarithms  of  a  few  numbers,  but  we 

need  the  logarithms  of  a  great  many  others.     All  possible 

numbers  must  lie  between  some  of  the  logarithms  now  ascer- 

taind.     The  numbers  between  1   and   10  must    have    their 

logarithms  between  0  and  1;  that  is,  the  logarithms  must  be 

fractions,  and  these  are  exprest  decimally  to  as  many  places 

as  desired,  the  difficulty  in  calculation  greatly  increasing  as 

the  number  of  places  is  increast.     Similarly,  as  the  numbers 

of  two  figures  lie  between  10  and  100,  their  logarithms  must 

lie  between  1  and  2;  that  is,   they  must  be  1   -f  a  decimal 

« 
fraction. 

5. — We  will  now  illustrate  the  properties  of  logarithms, 
confining  our  attention  to  the  single-figure  numbers  2,  3,  4,  5, 
6,  7,  8  and  9,  which  are  as  follows,  rounded  at  12  places : 


.301029  995  664 

In 

2 

.477  121254  720 

In 

3 

.602  059  991328 

In 

4 

.698  970  004  336 

In 

5 

.778  151250  384 

In 

6 

.845  098  040  014 

In 

7 

.903  089  986  992 

In 

8 

.954  242  509  439 

In 

9 

MuivTlPLICATION    BY    LOGARITHMS. 


6. — The  third  observation  in  Art.  1  leads  to  the  following 
rule  : 

The  sum  of  the  logarithms  of  several  numbers  is  the  logarithm 
of  their  product. 

2  111        .301029  995  664 

3  7il        .477  121254  720 
2x3=        (o    nl        .778  151250  384 

2  X  5=~ 

2  X  10=^ 
7.— In  the 

2 

5 
10 

2 

20 
logj 

nl 
nl 
nl 
nl 

7ll 

irith 

.301029  995  664 
.698  970  004  336 

1.000  000  000  000  (See  Art.  3) 

.301029  995  664 
1.301029  995  664 
ms  of  20,  200,  2000,  20.000,  200,000 

2,000,000,  etc.,  we  shall  find  the  same  decimal  part 
.301029  995  664,  {_ln  2)  preceded  by  the  figures  1,  2,  3, 
4,  5,  6,  etc.,  indicating  the  distance  from  the  units  place  of 
their  left-hand  figure,  or  the  number  of  zeroes  interpolated  to 
hold  that  position.  This  is  also  true  of  any  combination  of 
figures  ;  the  decimal  part  of  the  logarithm  is  the  same  what- 
ever their  place-value,  while  the  whole  number  prefixt  indi- 
cates the  place-value,  being  the  number  of  places  to  the  left 
of  units. 

Thus,  if  the  logarithm  of  2 .  378  is 

.376  211850  283,    then 

1.376  211850  283 

2.376  211850  283 

3.376  211850  283 

4.376  211850  283 

5.376  211850  283 
etc. 

8. — Where  the  number  is  less  than  unity  (a  decimal  frac- 
tion) the  characteristic  or  index  (the  prefixt  figure)  is 
negativ,  altho  the  decimal  (or  mantissa)  remains  positiv.  It 
is  usual  to  put  the  minus  sign   over  the  characteristic: 

.2378  {7il)  1.376  211850  283 

.02378  {7il)  2.376  211  850  283 

.002378  («/)  3.376  211850  283  • 

etc.  etc. 


23.78 

7ll 

237.8 

7ll 

2,378 

nl 

23,780 

nl 

237,800 

nl 

etc. 

4  The  Properties  of  Logarithms. 

Here  the  position  of  the  left-hand  figure  of  the  number  again 
determins  the  characteristic.  1  indicates  that  the  left-hand 
figure,  2,  is  in  the  Jirsi  place  to  the  right  of  the  unit  place  ;  2 
indicates  that  this  figure  is  in  the  second  place,  and  so  on. 
The  following  list  of  characteristics  will  show  that  the  left- 
hand  figure  of  the  combination,  by  its  location  to  the  right 
and  left  of  the  unit  figure,  determins  the  characteristic. 

Unit 

Places  0    000    000    00  0-     000    000    000 

Characteristics  9    876    543    21    0       1 2  3    45  6    78  9 

This  principle  saves  a  vast  amount  of  time  in  the  compu- 
tation of  logarithms,  and  also  in  their  application. 


9. — Since  division  is  the  converse  of  multiplication,  it  may 
be  performd  by  subtraction  as  that  is  by  addition. 

The  difference  of  the  logarithms  of  two   numbers   is  the 
logarithm  of  their  quotient. 

Required  the  quotient  of  6  -H  2. 

6    nl     .778  151250  384 

2     ''      .301029  995  664 

6/2    nl     .477  121254  720    In  3 
Required  the  value  of  >^  or  1  -f-  2 

1    nl     .000  000  000  000 

2     '•       .301029  995  664 

1/2     ;2/  1.698  970  004  336    In  .^ 
Required  the  value  of  Yi 

1    nl     .000  000  000  000 
3     ''      .477  121254  720 


1/3    w/  1.522  878  745  280    /;^  .3333 

.522  878  745  280  is  called  the  cologarithm  of   3  or  the 
logarithm  of  the  reciprocal  of  3. 

10. — Powers  of  numbers  are  found   by   multiplication. 
Let  it  be  required  to  find  the  third  power  of  2,  which  may  be 
written  2^  or  2x2x2.     By  the  process  first  shown 
2     nl     .301029  995  664 
2     "       .301029  995  664 

2_  "      .301029  995  664 

2  X  2  X  2  =  2^  nl     .903  089  986  992    In  8 


Roots  and  Powers. 

Required  the  square  (2d  power)  of  3 

3    nl     .477  121254  720 
3     "      .477  121254  720 


3'  nl     .954  242  509  440    In  9 
In  each  of  the  above  examples  it  would  have  been  simpler 
to  multiply  the  logarithm  by  the  exponent. 
2^  nl  (.301  029  995  664)  x  3  =  .903  089  986  992  In  8. 
3^  nl  (All  121  254  720)  x  2  =  .954  242  509  440  In  9. 
Therefore,  to  "raise"  a  number  to  a  certain  power,  we 
multiply  its  logarithm  by  the  exponent   and   then   find   the 
number  corresponding  to  the  product-logarithm. 

11. — The  second  power  is  usually  called  the  square,  and 
the  third  power  the  cube. 

12. — If  a  certain  number  is  a  power  of  another,  we  call 
the  latter  a  root  of  the  former.  Thus  if  2^  =  32,  we  may  say 
that  the  5th  root  of  32  is  2.     The  usual  way  of  expressing 

this  is  c-  / 

Y  32  =  2,  or 

32^  =  2. 
Using  the  latter  form  gives  a  symmetrical  list  of  ex- 
ponents and  their  meanings: 
a"    A  positiv  exponent  denotes  a  power 
a~**  A  negativ  exponent  denotes  the  reciprocal  of  a  power; 
ajr   A  fractional  exponent  denotes  a  root,  or  the  root  of  a  power; 
a*    The  exponent  ^  denotes  the  number  itself; 
a°    The  exponent  ^  denotes  unity.    ^ 

13. — As  roots  are  powers  with  fractional  exponents,  there- 
fore roots  are  found  (or  extracted)  by  dividing  logarithms 
insted  of  multiplying.  Thus  if  it  be  required  to  find  the  6th 
root  of  64,  we  take  (from  Colum  A  of  the  Table  of  Factors) 
the  logarithm  of  64,  and  divide  it  by  6. 
64^  nl  (1.806  179  973  984  /6)  ==  .301  029  995  664  In  2. 

Therefore  2  is  the  6th  root  of  the  number  64. 

14. — Such  an  exponent  as  f  may  require  explanation.  It 
signifies  the  third  power  of  the  fourth  root  or  the  fourth  root 
of  the  third  power. 

15. — Fractional  exponents  may  be  represented  as  decimal, 
insted  of  vulgar  fractions.  Thus  we  may  write  2*^^  insted  of 
2*  or  3*  for  3  .  In  fact,  that  is  what  most  logarithms  are: 
fractional  exponents  of  10,  exprest  decimally. 


The;  Properties  of  Logarithms. 


Tabi^es  of  Logarithms. 

16. — The  decimal  fractions  j^ch  constitute  that  part  of 
the  logarithm  requiring'  tU^>ilation  are  interminate ;  their 
values  may  be  computed  to  any  number  of  decimal  places.  If 
all  the  logarithms  in  a  certain  table  are  carried  to  5  deci- 
mal places,  it  is  called  a  5-place  table,  and  so  on.  Thus  the 
logarithm  of  2  has  been  computed,  with  great  labor,  to  20 
places  and  even  further. 

2    nl    .301029  995  663  981195  21  + 
In  a  4-place  table  this  would  be  rounded  off  to 

.3010; 
in  a    7-place,  .3010300; 
inalO-place,  .30102  99957; 

in  a  12-place,  .301029  995  664.  The  terminal  decimal  is 
never  quite  accurate,  but  is  nearer  than  either  the  next 
greater  or  the  next  less. 

17. — The  number  of  figures  in  the  numbers  for  which  the 
logarithms  are  given  must  also  be  considered.  The  tables 
most  in  use,  like  those  of  Vega,  Chambers  and  Babbage,  are 
of  five  figures  and  seven  places.  A  six-figure  table  would 
have  to  contain  ten  times  as  many  logarithms  and  occupy  ten 
times  the  space.  A  sixth  and  a  seventh  figure  may  be  obtaind 
from  them  by  interpolation.  The  United  States  Coast  Survey 
tables  (now  out  of  print)  are  five-figure  ten-places.  Nine 
figures  may  be  obtained  by  simple  proportion,  but  the  tenth 
is,  for  the  most  of  the  work,  unreliable.  Both  of  the  foregoing 
systems  give  auxiliary  tables  of  proportionate  parts,  or 
differences. 

18. — Peter  Gray  and  Anton  Steinhauser  have 
publisht  tables  of  24  and  20  places  respectivly,  but  the  plan 
for  extending  the  numbers  of  figures  is  quite  different  from 
the  simple  interpolation  above  referd  to.  They  both  procede 
by  subdividing  the  number  into  factors,  and  adding  together 
the  logarithms  of  those  factors. 

19. — All  logarithmic  calculations  end  with  the  ascertain- 
ment of  a  number  which  the  problem  calld  for.     The  more 


Tables  of  Logarithms.  7 

decimal  places  the  tables  give,  the  more  exact  the  resulting 
number,  or  answer,  will  be,  and  the  number  of  figures  in  the 
answer  can  never  be  more  than  the  number  of  places  in  the 
final  logarithm. 

20. — I  have  selected  twelve  figures  as  the  most  useful 
limit  for  the  accurate  computation  of  interest  problems,  that 
being  the  kind  for  which  the  work  is  specially  designd.  The 
logarithms  are  given  to  two  figures  and  thirteen  places,  the 
extra  place  insuring  the  accuracy  of  the  12th,  which  would 
otherwise  sometimes  be  1,  2  or  even  3  units  in  error,  thru  the 
roundings  being  preponderant  in  one  direction  or  the  other. 

21. — The  method  used  is  that  of  factoring,  it  being  pos- 
sible to  construct  the  logarithm  of  any  number  of  twelve 
figures  or  less  (900,000,000,000  in  all)  by  some  combina- 
tion of  the  584  logarithms  given  on  the  two  pages  of  the 
Table  of  Factors. 

Colum  A  contains  numbers  of  two  figures,  11  to  99,  and 
their  logarithms  to  thirteen  places. 

Colum  B  contains  the  logarithms  of  four-figure  numbers 
1.001  to  1.099,  each  beginning  with  1.0. . 

Colum  C  contains  the  logarithms  of  six-figure  numbers 
1.00001  to  1.00099,  each  beginning  with  1.000.  . . 

Colum  D,  1.0000001  to  1.0000099,  beginning  with  one 
and  five  zeroes. 

Colum  E,  1.000000001  to  1.000000099,  beginning  with 
one  and  seven  zeroes. 

Colum  F,  1.00000000001  to  1.00000000099,  beginning 
with  one  and  nine  zeroes. 

For  example,  opposit  34  in  the  table  we  find  : 

A     .531478  917  042,3    In     3.4 

B      .014  520  538  757,9    In     1.034 

C      .000147  635  027,3    In     1.00034 

D      .000  001476  598,7    In     1.0000034 

E      .000  000  014  766,0    In     1.000000034 

F      . 000  000  000  147,7    In     1 .  00000000034 

By  omitting  all  the  prefixt  zeroes,  the  printed  table  is  made 
very  compact,  each  line  containing  only  53  figures  insted  of 
78.  It  will  be  understood  hereafter  that  C  34,  for  example, 
means  the  number  1.00034,  and  F  34  means  1.00000000034. 


The  Properties  of  I^ogarithms. 


To  Find  the  Number  when  the  Logarithm  is  Given. 

22. — In  this  process  there  are  two  stages  :  first,  to  divide 
the  logarithm  into  a  number  of  partial  logarithms  among  those 
containd  in  the  T  F  (Table  of  Factors) ;  second,  to  multiply 
together  the  numbers  corresponding  to  these  logarithms.  Of 
course  the  decimal  part  only  of  the  logarithm  is  used  and  the 
number  has  the  position  of  its  units  figure  determind  from  the 
characteristic. 

23.— Let  the  logarithm  .753  797  472  366,5  be  one  which 
has  been  obtaind  as  the  result  of  an  operation,  and  the 
corresponding  number  be  required.  Search  in  Coluni  A  for 
the  highest  logarithm  which  does  not  exceed  the  given  one. 
This  is  found  to  be  .748  188  027  006,2,  which  stands  op- 
posit  56. 

Subtracting  from 753  797  472  366,5 

A  56        .748  188  027  006,2 

we  have  the  remainder 5  609  445  360,3 

This  is  smaller  than  any  logarithm 
in  Colum  A.  We  search  for  it  in 
Colum  B  and  find  opposit  13  pre- 
cisely the  same  figures 5  609  445  360.3 

These  two  logarithms  added  together  make  the  given  loga- 
rithm ;  hence  the  product  of  their  numbers  gives  the  number 
required. 

To  multiply  56  by  1.013 


56    ^ 

1013 

1013 

5 

5065 

56 

e 

6078 

56 
168 

r  °^ 

56728 

56728 

24. — This  process  may  be  greatly  simplified  as  follows, 
placing  the  figures  of  the  multiplier  in  vertical  order  at  the 

side: 

56  56 

66  13  X  5      065 

168  or  13  X  6        078 


56728  56728  , 

Notice  that  the  first  product  is  moved  two  colums  to  the  right 
of  the  multiplicand. 


To  Find  the  Number  when  the  Logarithm  is  Given.  9 


25. — We    will    now   take    a   little 

larger  logarithm 

and   continue   the   subtraction        A  56 

B13 

C26 


D29 


E58 


G65  + 


753  911  659  107,4 

748  188  027  006,2 

5  723  632  101,2 

5  609  445  360,3 

114  186  740,9 

112  901  888,7 

1  284  852,2 

1  259  452,2 


25  400,0 
25  189,1 


210,9 

208.5 
2,4 
2,4 


There  is  no  colum  G;  but  it  is  found  by  simply  taking  the 
first  two  figures  from  E.  It  may  be  either  55  or  56,  which 
may  make  the  thirteenth  figure  of  the  result  doutful,  but 
probably  not  the  twelfth. 


See  Note  1. 
See  Note  2. 

See  Note  3. 


See  Note  4. 


5600 
56 

168 


567280000 
113456 
3  4  0  3  6  8 


4  2  74928000 
* 
11.348550 
5106847 


507 


429 


97 
1.5. 

94 
70 
54 
28 
3 


5  6  7  4  2  9  17  15  2  6, 


lo  The  PropertiEvS  op  Logarithms. 

Note  1. — The  second  multiplication  jumps  its  right-hand 
figure  (6)  y^z^r  places  to  the  right,  which  may  be  markt  off 
by  four  zeroes,  or  four  dots. 

Note  2. — Having  extended  the  product  to  include  the 
13th  figure,  contraction  begins  in  this  multiplicand  ;  its  first 
figure  used  being  the  7th  (markt  *)  allowing  for  the  carrying 
from  the  8th.  Thus  the  starting  point  for  this  multiplication 
is  moved  six  places  back. 

Note  3. — The  multiplicand  need  no  longer  be  extended, 
as  has  been  done  at  successiv  stages  above,  but  remains  the 
same  to  the  end.  For  convenience,  dots  may  be  placed  in 
advance  under  the  first  figure  to  be  used  in  multiplication  in 
each  line.  j 

Note  4. — The  thirteenth  figures  are  added,  but  only  used 
for  carrying  to  the  twelfth.  In  this  example  the  total  of  the 
last  colum  is  31,  but  it  does  not  appear,  except  as  contribut- 
ing 3  to  the  next  colum. 

The  dot  below  a  figure  indicates  where  the  contracted 
multiplication  begins,  all  the  figures  to  the  right  being 
ignored,  except  as  to  their  carrying  power. 

25. — Another  example  in  which  there  is  no  suitable  loga- 
rithm in  A  and  w^e  must  begin  with  B. 

Required  the  number  for  log.     Oil  253  170  227 


To  Kind  the  Number  when  the  Logarithm  is  Given,  ii 


Formation  of  Number  from  Logarithm. 

Logarithm 

0 

1 

1 

2 

5 

3 

1 

7 

0 

1 

2 

7 

0 

A            — 

B  26 

1 

1 

1 

4 

7 

3 

6 

0 

7 

7 

5 

8 

1 

0 

5 

8 

0 

9 

3 

5 

1 

2 

C  24 

1 

0 

4 

2 

1 

8 

1 

7 

0 

0 

1 

5 

9 

1 

1 

8 

1 

2 

D  36 

1 

5 

6 

3 

4 

5 

7 

3 

2 

7 

7 

2 

3 

9 

E  63 

— 

— 

2 

7 

3 

6 

0 

5 

3 

6 

3 

4 

F  83 

3 

6 

0 

5 

2 

9 

G  67  ^^^'^ 

2 

9 

A            — 

B 
26 

1 

0 

2 

6 

1 

0 

2 

6 

• 

C     2 

2 

0 

5 

2 

4 

4 

1 

0 

4 

1 

0 

2 

6 

2 

4 

6 

2 

4 

• 

D     3 

3 

0 

7 

8 

7 

3 

9 

6 

• 

6 

1 

5 

7 

4 

8 

1 

0 

2 

6 

2 

4 

9 

9 

3 

4 

4 

8 

7 

E     6 

• 

6 

1 

5 

7 

5 

3 

• 

3 

0 

7 

9 

F     8 

• 

8 

2 

1 

3 

• 

3 

1 

G     6 

• 

6 

7 

1 

1 

0 

2 

6^ 

2_ 

5 

0^ 

01 

0 

0^ 

^ 

0^ 

In  this  example  we  illustrate  the  procedure  when  B 
furnishes  the  first  logarithm.  It  also  shows  the  convenience 
of  using  paper  ruled  for  the  purpose. 


12  The  Properties  of  Logarithms. 

26. — In  order  to  set  down  the  partial  products  without 
hesitation,  remember  the  numbers  2,  4,  6. 
In  multiplying  by  B 

the  first  figure  of  the  product  moves  two  places  to  the  right. 
In  multiplying  by  C 

the  first  figure  of  the  product  movers/our  places  to  the  right. 
In  multiplying  by  D 

the  first  figure  of  the  7nultiplica?td  moves  six  places  to  the  left. 

27. — The  following  rule  may  now  be  formulated  for  this 
process: 

Rule. — 1.  By  successiv  subtractions  separate  the  given 
logarithm  into  a  series  of  partial  logarithms  found  in  the 
colums  of  the  T  F,  setting  opposit  each  its  letter  and  number. 

2.  By  successiv  multiplications  find  the  product  of  all  the 
numbers  thus  found,  allowing,  in  the  placing  of  the  partial 
products,  for  the  prefixt  1  and  zeroes. 

28.  The  work  may  be  made  to  occupy  fewer  lines  by 
setting  down  the  factors  E,  F  and  G  as  one  number  at  the  top, 
multiplying  it  by  A  and  incorporating  it  thereafter  as  one 
multiplicand  with  the  preceding  figures.  The  result  will  not 
be  affected.  Let  the  factors  be,  as  before:  A  56  B  13  C  26 
D29    E58    F48     G  55. 

E  F  G 

584855 


^^^  2924275 

350913 

5600000327519 

B13       56000003275 

16800000  9  82 

5672800331776 

C26        1134560066 

340368029 

567427525987  1 

D29  11348551 

5106848 

567429171  5  afiXL 


To  Find  the  Number  when  the  Logarithm  is  Given.   13 

29. — Required  the  number  whose  logarithm  is  .5  or  >^. 

.500  000  000  000.0 
A  31  491  361  693  834,3 


B  20 

8  638  306  165,7 
8  600  171  761,9 

C  08 

38  134  403,8 
34  742  168,9 

D  78 

3  392  234,9 
3  387  483,7 

E  10 

4  751,2 
4  342,9 

F  94 
G  03 

408,3 
408,2 

0,1 

The  resulting  factors 

A  31    B  20    C  08    D  78    E  10    F  94    G  03 
when  combined  produce  the  result  3.16227766017. 

30. — The  multiplication  illustrates  how  zeroes  are  treated 
when  they  occur  in  the  multipliers. 

31. — The  result  is  the  square  root  of  10,  to  12  places,  as  may- 
be demonstrated  by  multiplying  3.16227766017  by  itself. 
Method  by  Multipi^es. 
32. — In  order  to  facilitate  the  multiplication  of  the  factors, 
A,  B,  C,  etc.,  Mr.  A.  S.  Little,  of  St.  Louis,  has  devised  a  Table 
of  Multiples,  giving  the  product  of  each  number  from  1  to  9 
by  every  number  from  2  to  99.  (See  page  35.)  Thus  the 
multiples  of  89  read  in  one  line  as  follows  : 

123456789 
089    178    267    356    445    534    623    712    801 
Then,  if  it  be  desired,  for  example,  to  multiply  68792341 
by  89,  we  would  select  from  the  above  table 
under  6  5  3  4 

8  712 

7       623 

9  801 

2  178 

3  267 

4  356 

1    089 

6122518349 


14  .Thk  Properties  of  I^ogarithms. 

We  have  thus  multiplied  each  figure  of  the  multiplicand 
by  both  figures  of  the  multiplier,  setting  down  each  partial 
product  unhesitatingly. 

33.  — The  work  may  be  made  more  compact  by  piling  the 
partial  products  like  bricks,  using  only  three  lines: 
5  3  4.801,356, 
7  12,178,089 
6  2  3,2  6  7, 

6122518349 
34. — Three  figures  must  be  set   down   for  each  partial 
product,  even  if  the  first  be  a  zero. 

35. — To  use  this  method  in  combining  the  factors  of  a 
number,  the  letters  A,  B,  C,  etc.,  are  written  above  alternate 
figure  spaces,  which  is  facilitated  by  the  use  of  paper  properly 
ruled.  Then  the  first  partial  product  under  each  letter  is 
placed  with  its  middle  figure  under  that  letter  at  the  top. 

36. — The  following  is  an  example  of  a  combination  al 
ready  performd  in  another  form  : 

A  B  C  D  E  F  G 
A  56   1  584855 

2  8  0,448 
4  4  8,280 
2  2  4,2  8 


56       327519 

B  13     0  6  5      0  3  9,0  6 

078      0  2  6,2 

091 

5672800331778 
C26       13  0.052,000, 
15  6,208,078 
18  2.0  0  0,0  8 

5674275259864 

D  29         14  5.116,15 

17  405  8,1 

2  0  3.203 

5  67429171526 
37. — Mr.  Little  has  also  suggested  a  process  for  verifying 
a  numerical  result  by  using  a  different  set  of  factors  in  a 
second  operation. 

38. — Required  the  number  corresponding  to 
.305  773  384  163.0 


To  Find  the  Number  when  the  Logarithm  is  Given.   15 

The  factors  are  A  20  B  10  C  97  D  21  E  94  F  94  G  33. 
The  number  is  2.02195383809. 

In  order  to  check  the  result  and  make  sure  of  perfect  ac- 
curacy, we  may  solve  the  problem  a  second  time,  using  two 
subtrahends  from  A.  The  first  subtraction  may  be  of  any 
suitable  number  ;  11  is  found  to  give  the  greatest  facility. 

305  773  384  163,0 

A  11  041  392  685  158.2 

264  380  699  004,8 

A  18  255  272  505  103,3 

9  108  193  901,5 

B  21  9  025  742  086,9 

82  451  814,6 

C  18  78  165  972,0 

4  285  842,6 
D  98  4  256  065,1 

29  777,5 
E  68  29  532.0 

245,5 
F  56  243.2 

2.3 
G  53  2.3 

39. — The  remainder  of  the  operation  may  be  by  either  method: 
A     B    C    D   E     F    G  A     B    C     D   E    F    G 

A  18  685653A18  685653 

548522  548522 

18  1234175       18       1234175 
A  11   18 123418  A  11   1^ 123418 

19  8      135759        198      135759 
B21    396      2715    B21   021      02  1,15 

198      136  18  9      0  6  3,1 

2021580138610        1 68      105 

C18       202158014  202  1580138610 

16172  6  411  C  18     0  3  6,0  1  8,0  0  0,1 

2021944023035  0  0  0.090.018 

D98        18197496        0  3  6.14  4.0  5 

1617555  2021944023034 

202196383809  D  98        19  6  0  9  8,39 

0  0  0.8  8  2.0 

♦  196392 


202196383809 
In  this  example  the  first  method  appears  to  be  preferable, 
especially  in  the  earlier  part. 


1 6  The  Properties  op  Logarithms. 


To  Form  the  Logarithm  of  a  Number. 

40.  — This  consists  in  two  processes  :  first,  the  number  is 
separated  into  a  series  of  factors  corresponding  to  the  six 
colums  of  the  thirteen-place  table  ;  second,  the  logarithms  of 
these  factors  are  copied  from  the  table  and  added  together. 

41. — The  factoring  is  effected  by  a  progressiv  division, 
the  divivSor  receiving  successivly  more  and  more  of  the  figures 
of  the  number. 

42. — To  illustrate  this  division  we  will  assume  a  number 
in  which  the  division  will  be  soon  completed. 

To  find  the  logarithmic  factors.  A,  B,  C,  etc.,  of  5.6728. 
First  extend  the  number  to  12  places,  567  280  000  000.  The 
first  factor  A  is  always  the  first  two  figures  of  the  number 
itself.  A  56)56  7  2  80  000  000  (1 .  013  B 

56 
72 
56 

168 
168 


It  will  readily  be  seen  that  one  56  might  have  been  omitted. 
A  56)7  280  000  000  (B  13 
56 

168 
168 

Turning  then  to  the  Table  we  have  only  to  set  down  the 
logarithms  of  these  two  factors  : 

A  56  nl    748  188  027  006,2 
B   13  nl        5  609  445  360,3 

56728     nl    753  797  472  366  5 
B  13  may  be  regarded  as  an  abbreviation  of  L.013. 

43.— We  will  now  give  an  example  where  a  second  divisor, 
at  least,  is  required. 

A  56)  7  4  2  9  1  7  1  5  2  6  (B  13 
56 


182 
168 


A  B  56  728  )  1  4 


To  Form  the  Logarithm  op  a  Number.  17 

The  second  divisor  is  the  product  of  A  and  B.     It  might 
be  obtaind  in  either  of  three  ways. 

1.013  =  56728 


multiplication 

56  X 

addition 

5Q 

+      56 

+      168 

56728 
But  the  easiest  way  is 

by  subtraction  56742  five  figures  of  the  number 

—       14  the  remainder 
56728 
This  is  the  proper  method  for  forming  all  divisors  after 
the  first ;  subtract  the  remainder  from  the  original  number  so 
far  as  used. 

44. — We  resume  the  division,  bringing  down  /our  more 
figures,  to  the  ninth  inclusiv. 

AB)56728        )14  9  171526(C26 
113456 
357155 
340368 


ABC)56742  7,5)     *1678726(D29 

1134855 

543871 

510685 

5  67  42  9,174-  3  3  1  8  6  (E  58 

28371 

4815 

4539 


2  7  6  (F  48,7 

227 

49 

45 

4 

The  third  divisor  A  B  C  is  also  formd  by  subtracting  from 

the  number  5674291715 

*  the  remainder     16  7  8  7 

5674274928 


1 8  Thb  Properties  of  Logarithms. 

As  only  six  figures  are  needed  for  the  divisor  and  one  for 
carrying,  this  is  rounded  up  to      5  6  7  4  2  7,5 

The  fourth  divisor  is  practically  the  number  itself  so  far 
as  needed,  and  this  lasts  to  the  end. 

45. — The  entire  process  is  now  repeated,  but  for  greater 
accuracy  in  the  twelfth  figure  we  will  divide  out  to  the 
thirteenth.     . 

A56)  7  429  17  15  26,0  (B  13 
56 


A  B  56  728) 


182 
168 

149171 
113  4  5  6 


(C26 


357155 
340368 


A  B  C  56  742  749  28) 
[Contracted  division  begins  here] 


56  742  92) 


1  6  7  8  7  2  6,0  (D  29 
113485  5,0 


5  4  3  8  7  1,0 
5  106  84,7 


33  186,3  (E58 
2837  1,4 

4  8  1  4,9 
45  3  9,4 


2  7  5,5  (F48 
2  2  7.0 


485 
454 


(G55 


31 

28 


It  remains  only  to  add  together  the  logarithms  : 
A  56  {nl)    748  188  027  006,2 


B  13  " 

5  609  445  360,3 

C  26  •' 

112  901  888,7 

D29  " 

1  259  452,2 

E58  " 

25  189,1 

F  48  " 

208,5 

G55  '* 

2.4 

567  429  171  526  («/) 

753  911  659,107 

To  Form  the  Logarithm  of  a  Number.  19 

46. — The  figures  in  the  last  colum  are  only  usg^  for  car- 
rying to  the  twelfth,  which  otherwise  would  give  Stnsted  of  7. 

47.  — We  may  now  formulate  the  following  rule  for  finding 
the  logarithm: 

Rule.— 1.  Make  the  number  to  13  figures,  by  adding 
cifers  or  cutting  off  decimals. 

2.  Cut  off  the  two  left-hand  figures  by  a  curve,  giving  A. 

3.  Divide  the  next  three  figures  by  A,  giving  the  two 
figures  of  B,  and  a  remainder. 

4.  Form  the  second  divisor  A  B,  by  subtracting  the  re- 
mainder from  the  first  five  figures  of  the  number. 

6.  Bring  down  four  more  figures  to  the  remainder  and 
divide  by  A  B,  giving  the  two  figures  of  C  and  a  remainder. 

6.  Form  the  third  (and  last)  divisor  A  B  C  by  subtracting 
the  remainder  from  ten  figures  of  the  number. 

7.  Divide  the  remaining  figures  by  the  third  divisor.  As 
there  are  ten  figures  in  the  divisor  and  only  eight  in  the  divi- 
dend, contraction  begins  immediately.  Having  obtaind  the 
figures  of  D,  the  divisor  for  E,  F  and  G  is  simply  the  number 
itself  contracted. 

8.  Write  down  the  logarithms  of  A,  B,  C,  D,  E  and  F, 
obtaind  from  the  several  colums  of  T  F;  also  that  of  G,  being 
the  first  two  figures  of  E.  The  sum  will  be  the  logarithm,  the 
thirteenth  figure  being  used  for  carrying  only. 

48. — It  is  advisable  to  make  all  logarithmic  computations 
on  paper  ruled  with  thirteen  down-lines,  every  third  being 
darker.     A  specimen  is  given  on  page  36. 

49.  A  few  examples  for  practis  are  given  below  with  the 
factors  and  the  solution: 

5674  =  A  56    B  13    C  21     D  15    E  35    F  42    G  70 
log.  5674  =  3.753  889  331  458 
38.8586468578  =  A  38    B  22    C  58    D  31     E  39    F  02    G  25 

log.  do.  =  1 .589  487  673  453 

3.1415926535898  +  =  A  31     B  13    C  41     D  16    E  33    F  11    G  91 
log.  do.  =.497  149  872  694 

(This  number  is  the  ratio  of  the  circumference  of  a  circle  toils  diameter.) 
1.02625  =  B  26    C  24    D  36    E  63    F  83 
log.      do.      =  .011  253  170  127 
This  number  begins  with  an  expression  of  the  form  B  (1.026),  hence 
no  division  by  A  occurs.     1026  is  the  first  divisor. 


20  The  Properties  of  I^ogarithms. 


B    1026)  2  5  0  0 
2052 

C24 

4480 

4104 

B  C     102624624)    3  7  6  0  0  0.0 
3  0  7  8  7  3,9 

D36 

6  8  1  2  6,1 

6  1574,8 

102625) 

6  55  1,3 

6  1  5  7,5 

3  9  3,8 

3  0  7,9 

K63 

8  5,9 

8  2,1 

F83 

3,8 
3,1 

7 

G70 

B  26 
C24 
D36 
E63 
F83 
G70 

Oil  147  360  775,8 

104  218  170,0 

1563  457,3 

27  360,6 

360,5 

3,0 

Oil  253  170  127 

This  result  will  be  found  also  in  the  Table  of  Interest-Ratios,  but 
even  more  extended. 


21 


Logarithms  to  Less  than  12  Places. 

60. — The  T  F  may  be  cut  down  to  any  lower  number  of 
places.  In  the  example  in  Art.  45  it  may  be  required  to  give 
9  places  only,  the  tenth  being  used  for  carrying.  We  cut 
down  the  original  logarithm  to  ten  figures,  with  a  comma  after 
the  ninth  and  it  becomes 

753  911  659,1 
A  56     748  188  027.0 


5  723  632,1 

B  13 

5  609  445,4 

114  186,7 

C  26 

112  901.9 

1  284,8 

D  29 

1  259,5 

25,3 

B58 

25,2 

F  24 

1 

A 

56 

B  1 

5  6 

3 

168 

567280000 

C  2 

113456 

6 

3  4036,8 

56742749  2,8 

D  2 

1  1  3  4,9 

9 

5  10,7 

E  5 

2  8,4 

8 

4,5 

F  2 

1 

56742917  1,4 

The  number  is  slightly  in  error  in  its  tenth  place,  but 
correct  to  the  ninth. 

51. — If  a  table  of  factors  for  18  or  some  other  number  of 
places  should  hereafter  be  prepared,  the  methods  which  have 
been  explaind  would  be  applicable. 


22  The  Properties  of  Logarithms. 


Multiplying  Up. 

52. — Mr.  Edward  S.  Thomas,  of  Cincinnati,  has  suggested 
another  method  for  obtaining  the  factors  of  the  number  in 
forming  its  logarithm. 


53. — It  procedes  by  multiplication  insted  of  division,  the 
latter  operation  being  notabh^-  the  more  laborious.  The  num- 
ber, at  first  taken  as  a  decimal  less  than  1,  is  successivly  mul- 
tiplied up  to  produce  1.000,000,000,0  and  these  multipliers 
are  the  A,  B,  C,  D,  E,  F  and  G,  whose  logarithms  added  to- 
gether make  the  cologarithm,  from  which  the  logarithm  is 
easily  obtaind. 


54. — A  is  a  number  of  two  figures,  a  little  less  than  the 
reciprocal  of  the  number,  which  will  be  calld  the  sub-recipro- 
cal of  its  two  initial  figures.  A  Table  of  Sub- Reciprocals  is 
given  on  page  33.  The  number  multiplied  by  A  will  always 
give  a  product  beginning  with  9.  B  is  always  the  arithmetical 
complement  of  the  two  figures  following  tlve  nine,  or  the  re- 
mainder obtaind  by  subtracting  those  two  figures  from  99. 
Multiplication  by  B  will  usually  give  a  result  beginning  with 
999.  C  is  the  next  complement  and  gives  5  9's,  999,99.  D 
similarly  brings  999,999,9  >{^  *  ^  *  >K  H^  ^  *  .  No  further  multipli- 
cation is  necessary,  when  D  has  been  used  ;  the  six  figures  in 
the  places  of  the  stars  are  the  complements  of  E,  F  and  G. 


55. — To  illustrate,  let  it  be  required  to  obtain  the  loga- 
rithm to  the  12th  place  of  3.14  159  265  359  0.  The  object  is 
to  multiply  .314  159  265  -859  up  to  1 .000  000  000  000  0.  The 
first  step  is  to  find  the  sub-reciprocal  of  31,  or  A.  Turning  to 
the  Table  of  Sub-reciprocals,  opposit  81  we  find  31,  by  which 
we  multiply. 


Multiplying  Up. 


23 


A  31 


99  —  73  =  26 

B  26  is  therefore 
the  next  multipli- 
er; dropping  the 
last  two  figures 

(99—21)      C  78 


(99—43)      D  56 


(99-47) 

(99-03) 

(100-77) 


E  52 
F  96 
G23 


.3141592653590 

.9424777960770 
31415  92653  5  9 
.9  738937226129   One  9  has  been  secared 


194778744523 
58433623357 


999214959400  9 

6994504716 

799  3  719  6  8 


Three  nines  secured 


99999434706  9  3  Five  nines 
49999718 
5999966 


.9999999470377    Seven  nines 
52 
9  6 
23 


A31;^/  .4  913616938343 

B26  111473607758 

C78  3386176522 

D56  24320423 

E52  225833 

F  96  4  16  9 

G  23   _j 1^ 

colog.  0.5  02850127306 

log.  1.4  97149872694 


56. — It  may  happen,  in  the  course  of  multiplication,  that 
the  complement  of  the  figures  following  the  9  does  not  suffice 
to  secure  two  nines  more.  In  this  case,  another  supplementary 
multiplication  must  take  place.  This  occurs  in  the  following 
example,  which  has  alredy  been  solvd. 


24  The  Propkrties  of  I,ogarithms. 

57. — Required  the  logarithm  of  the  number 
567  429  171  526. 

In  this  example  the  C  multiplication  also   requires   an 
additional  figure.     This  seldom  occurs. 


.567  429171526  0 
A  17       .397  200  420  068  2 

.964  629  591594  2 

B  35  28  938  887  747  8 

4  823  147  958  0 

.998  391627  300  0 
"  01  998  391  627  3 

.999  390  018  927  3 
C  60  599  634  0114 

.999  989  652  938  7 
"  01  9  999  896  5 

.999  999  652  835  2 

D  03      299  999  9 

.999  999  952  835  1 
47  164  9 


{ 


A  17 

230  448  921  378  3 

B  35 

14  940  349  792  9 

01 

434  077  479  3 

C  60 

260  498  547  4 

01 

4  342  923  1 

D  03 

130  288  3 

B  47 

20  411  8 

F  16 

69  5 

G  49 

21 

.246  088  340  892  7 

.753  911659  107  3 

As  the  multiplication  by  35  brings  only  998  insted  of  999, 
we  multiply  again  by  B  01,  which  brings  it  up. 


MuivTiPLYiNG  Up.  25 

58. — In  the  next  example  there  is  a  large  defect  in  B, 
which  requires  an  additional  multiplication  by  7. 

110  175 


A  83    881  400  (83,  subreciprocal  of  11) 
33  052  5 


914  452  5 
B  85    73156  200 
4  572  262  5 


B  07 
C  87 


999  995  469  056,9 

D   45  3  999  981,9 

499  997,7 


992  180  962  5 
6  945  266  737,5 

999  126  229  237.5 

799  300  983,4 

69  938  836,0 

999  999  969  036,5 
30  963,5 


A  83 

919  078  092  376,1 

B  85 

35  429  738  184,5 

B  07 

3  029  4*70  553,6 

C  87 

377  670  935,8 

D  45 

1  954  320^8 

K  30 

13  028,8 

F  96 

416,9 

G  35 

1,5 

957  916  940  818  0 
042  083  059  182  0 


The  number  11075  was  purposely  selected,  very  slightly 
in  excess  of  the  highest  number  in  colum  B,  so  as  to  produce 
the  shortage  of  7. 


26  The  Properties  of  I^ogarithms. 

59. — Little's  Table  of  Multipliers  may  be  used  in  the 
multiplication,  as  in  the  following  example.  It  will  be  found 
•that  the  logarithm  when  computed  has  the  same  figures  as  the 
number  itself;  a  remarkable  peculiarity  which  no  other  com- 
bination of  figures  can  possess. 

.  137  128  857  423  9 


A 

71 

0  710  715  682  846  4 

213  142  355  142  0 

49  756  849  721  3 

.  973  614  887  709  7 

B 

26 

23  415  620  818  2 

1  820  262  080  0 

078  104  182  2 

.  998  928  874  790  1 

tt 

01 

998  928  874  8 

.  999  927  803  664  9 

C 

07 

69  994  946  3 

.  999  997  798  611  2 

D 

22 

1  981  981  8 

198  198  0 

19  815  4 

.  999  999  998  606  4 

01  393  6 

E  F  G 

A 

71 

851  258  348  719  1 

B 

26 

11 147  360  775  8 

B 

01 

434  077  479  3 

C 

07 

30  399  549  8 

D 

22 

955  446  8 

E 

01 

434  3 

F 

39 

169  4 

G 

36 

16 

862  871  142  576  1 

.137  128  857  423  9 
which  is  the  log.  of  1.371  288  574  239 


27 

Signs  of  the  Characteristic. 

60 — We  have  seen  (Art.  8)  that  while  the  decimal  part 
of  the  logarithm  is  always  positiv,  the  characteristic  is  often 
negativ  and  has  the  minus  sign  above  it. 

61. — In  adding  together  several  logarithms  with  different 
signs,  the  positivs  and  the  negativs  must  be  added  separately; 
the  less  sum  must  be  subtracted  from  the  greater,  and  the 
remainder  has  the  sign  of  the  greater  sum.  The  carrying 
from  the  decimal  part  counts  with  the  positivs. 


34    7il. 

1.531478  917  042,3 

2900      " 

3.462  397  997  899,0 

.73      '' 

1.863  322  860120,5 

.056      ♦' 

2.748  188  027  006.2 

The  sum  of  the 

decimals  is.  .  . . 

2.605  387  802  068,0 

The  positivs  are 

1. 

and 

3 

Total 

+  6 

The  negativs  are    1 

, 

2 

—  3 

Sum  of  the  logarithms  +3.605  387  802  068,0 

The  decimal  point  in  the  result  must  follow  the  fourth 
figure,  as  indicated  by  the  characteristic  3. 

62. — In  subtracting  one  logarithm  from   another,    when 
the  decimal  of  the  subtrahend  is  the  greater,   and  a  unit^ 
"  borrowed,"  the  unit  is  considered  as  one  more  negativ: '^ut   ^CT^'^'y''  * 
the  total  characteristic  changes  its  signs  from  plus  to  minus  or    /Tv^^^ 
from  minus  to  plus.  . 

290  2  462  397  997  899,0  <^^^^=-^  ^ 

.0058  3.763  427  993  562,9 
Operate  on  the  deci- 
mals only 0 .  462  397  997  899,0 

0.763  427  993  562,9 

1.698  970  004  336,1 
Negativ  from  subtrahend  3 
Total  negativ 4 


Sign  changed +4  Y^^^"^^^ 

From  minuend  ...    -h  2  ) 


6 .  698  970  004  336,1  In    5  000  000^ 


y 


28  The    PrOPERTIBS    of   lyOGARlTHMS. 

63. — To  multiply  a  logarithm  having  a  negativ  character- 
istic (in  order  to  obtain  a  power  of  a  decimal) ,  multiply  the 
decimal  part  and  the  characteristic  separately  and  add  the  two 
together: 

2.301029  995  664.0     x  5 

Decimal  part _  1 .  505  149  978  320,0 

Characteristic 10. 

9.505  149  978  320,0 
Therefore  the  5th  power  of  .02  is  .000  000  003  2. 

64.  — To  divide  a  logarithm  having  a  negativ  characteristic, 
(for  the  extraction  of  a  root;)  if  the  characteristic  is  exactly 
divisible,  divide  the  decimal  part  and  the  characteristic 
separately: 

12.690  196  080  028,5 -f- 6 
2.115  032  680  004,7 

But  if  the  characteristic  be  not  so  divisible,  add  to  it  a 
negativ  quantity,  which  will  make  it  divisible,  and  prefix  to  the 
decimal  part  in  compensation  an  equal  quantity  positiv. 

12.690  196  080  028,5 -f- 5 
Add  3  15. 

Add  3  3.690196  080  028.5 

Quotient  3 .  738  039  216  005,7 

Different  Bases. 

65. — Ten  is  the  base  of  the  logarithmic  system  which  we 
have  been  explaining;  it  is  the  most  useful  of  all  systems, 
because  ten  is  also  the  base  of  our  numerical  system.  These 
are  usually  calld  common  or  vulgar,  or  Briggsian  logarithms, 
but  decimal  logarithms  would  seem  more  appropriate. 

66. — Any  number  might  form  the  base  of  a  system  of  lo- 
garithms, but  the  only  other  in  actual  use  is  one  known  as 
the  "natural"  system,  having  for  its  base  the  number 
2. 718281828459 -h  which  is  the  sum  of  the  series 

1       J_  1  1  1 

■^  "^  i  "^  1x2  "^  1x2x3  "^  1x2x3x4  "^  1x2x3x4x5^^^' 

This  is  only  used  in  theoretical  inquiries,  and  is  seldom  of 
utility  to  the  accountant. 


PART  II. 

TABLES 

FOR  OBTAINING 

Logarithms  and  Antilogarithms 

TO  12  PLACES  OF  decimals 


30 


TABLE  OF  FACTORS 


(V 

a 

01 
02 
03 
04 

05 
06 
07 

08 
09 

10 
11 
12 
13 
14 

15 
16 
17 

18 
19 

20 
21 
22 
23 
24 

25 
^26 

27 
28 
29 

30 
31 
32 
33 
34 

35 
36 
37 
38 
39 

40 
41 
42 
43 
44 

45 
46 
47 
48 
49 

A 

*  .  * 

' 

041  392  685  158,2 
079  181  246  047,6 
113  943  352  306,8 
146  128  035  678,2 

176  091  259  055,7 
204  119  982  655,9 
230  448  921  378,3 
255  272  505  103,3 
278  753  600  952,8 

301  029  995  664,0 
322  219  294  733,9 
342  422  680  822,2 
361  727  836  017,6 
380  211  241  711,6 

397  940  008  672,0 
414  973  347  970,8 
431  363  764  159,0 
447  158  031  342,2 
462  397  997  899,0 

477  121  254  719,7 
491  361  693  834,3 
505  149  978  319,9 
518  513  939  877,9 
531  478  917  042,3 

544  068  044  350,3 
556  302  500  767,3 
568  201  724  067,0 
579  783  596  616,8 
591  064  607  026,5 

602  059  991  328,0 
612  783  856  719,7 
623  249  290  397,9 
633  468  455  579,6 
643  452  676  486,2 

653  212  513  775,3 
662  757  831  681,6 
672  097  857  935,7 
681  241  237  375,6 
690  196  080  028,5 

B 

1.0*  H« 


434  077  479,3 

867  721  531,2 

1  300  933  020,4 

1  733  712  809,0 

2  166  061  756,5 

2  597  980  719,9 

3  029  470  553,6 
3  460  532  109,5 
3  891  166  236,9 


4  321  373  782,6 

4  751  155  591,0 

5  180  512  503,8 

5  609  445  360,3 

6  037  954  997,3 

6  466  042  249,2 

6  893  707  947,9 

7  320  952  922,7 

7  747  778  000,7 

8  174  184  006,4 


8  600  171  761,9 

9  025  742  086,9 
9  450  895  798,7 
9  875  633  712,2 

10  299  956  639,8 

10  723  865  391,8 

11  147  360  775,8 
11  570  443  597,3 

11  993  114  659,3 

12  415  374  762,4 


12  837  224  705,2 

13  258  665  283,5 

13  679  697  291,2 

14  100  321  519,6 
14  520  538  757,9 

14  940  349  792,9 

15  359  755  409,2 

15  778  756  389,0 
16*197  353  512,4 

16  615  547  557,2 


17  033  339  298,8 
17  450  729  510,5 

17  867  718  963,5 

18  284  308  426,5 

18  700  498  666,2 

19  116  290  447,1 
19  531  684  531,3 

19  946  681  678,8 

20  361  282  647,7 
20  775  488  193,6 


C 
1.000+* 


4  342  923,1 

8  685  802.8 

13  028  639,0 

17  371  431,8 

21  714  181,2 

26  056  887,2 

30  399  549,8 

34  742  168,9 

39  084  744,6 


43  427  276,9 

47  769  765,7 

52  112  211,2 

56  454  613,2 

60  796  971,8 

65  139  287,0 

69  481  558,7 

73  823  7^7,1 

78  165  972,0 

82  508  113,5 


86  850  211,6 

91  192  266,3 

95  534  277,6 

99  876  245,5 

104  218  170,0 

108  560  051,0 

112  901  888,7 

117  243  682,9 

121  585  433,8 

125  927  141,2 

130  268  805,2 

134  610  425,9 

138  952  003,1 

143  293  536,9 

147  635  027,3 

151  976  474,3 

156  317  878,0 

160  659  238,2 

165  000  555,0 

169  341  828,4 


173  683  058,5 

178  024  245,1 

182  365  388,3 

186  706  488,2 

191  047  544,7 

195  388  557,7 

199  729  527,4 

204  070  453,7 

208  411  336,6 

212  752  176,1 


D 

1.00000** 


43  429,4 

86  858,9 

130  288,3 

173  717,8 

217  147,2 

260  576,6 

304  006,0 

347  435,4 

390  864,9 


434  294,3 

477  723,7 

521  153,1 

564  582,5 

608  011,8 

651  441,2 
694  870,6 
738  300,0 
729.4 
158,7 


781 
825 


868  588,1 
912  017,5 
955  446,8 
998  876,2 
1  042  305,5 

1  085  734,8 

1  129  164,2 

1  172  593,5 

1  216  022,8 

1  259  452,2 


302 
346 
389 
433 
476 


881,5 
310,8 
740,1 
169,4 
598,7 


520  028,0 
563  457,3 
606  886,6 
650  315,9 
693  745,2 


737  174,5 
780  603,7 
824  033,0 
867  462,3 
910  891,5 


1  954  320,8 

1  997  750,0 

2  041  179,3 
2  084  608,5 
2  128  037,7 


^H 

o 

E 

F 

a  • 

3 

1.0'** 

1.0%* 

^ 

434,3 

004,3 

01 

868,6 

008,7 

02 

1  302,9 

013,0 

03 

1  737,2 

017,4 

04 

2  171,5 

021,7 

05 

2  605,8 

026,1 

06 

3  040,1 

030,4 

07 

3  474,4 

034,7 

08 

3-908,7 

039,1 

09 

4  342,9 

043,4 

10 

4  777,2 

047,8 

11 

5  211,5 

052,1 

12 

5  645,8 

056,5 

13 

6  080,1 

060,8 

14 

6  514,4 

065,1 

15 

6  948,7 

069,5 

16 

7  383,0 

073,8 

17 

7  817,3 

078,2 

18 

8  251,6 

082,5 
086.9 

19 

20 

8  685,9 

9  120,2 

091,2 

21 

9  554,5 

095,5 

22 

9  988,8 

099,9 

23 

10  423,1 

104,2 

24 

10  857,4 

108,6 

25 

11  291,7 

112,9 

26 

11  726,0 

117,3 

27 

12  160,2 

121,6 

28 

12  594,5 

125,9 
130,3 

29 
30 

13  028,8 

13  463,1 

134,6 

31 

13  897,4 

139,0 

32 

14  331,7 

143,3 

33 

14  766,0 

147,7 

34 

15  200,3 

152,0 

35 

15  634,6 

156,3 

36 

16  068,9 

160,7 

37 

16  503,2 

165,0 

38 

16  937,5 

169,4 
173,7 

39 
40 

17  371,8 

17  806,1 

178,1 

41 

18  240,4 

182,4 

42 

18  674,7 

186,7 

43 

19  109,0 

191,1 

44 

19  543,3 

195,4 

45 

19  977,5 

199,8 

46 

20  411,8 

204,1 

47 

20  846,1 

208,5 

48 

21  280,4 

212,8 

49 

TABLE  OF 

FACTORS— Continued 

31 

HI 

A 

B 

C 

D 

E 

F 

s 

50 

*  •  * 

1.0** 

1.000** 

1.00000** 

1.0%* 

1.0%* 
217,1 

S 
50 

698  970  004  336,0 

21  189  299  069,9 

217  092  972,2 

2  171  467,0 

21  714,7 

51 

707  570  176  097,9 

21  602  716  028,2 

221  433  725,0 

2  214  896,2 

22  149,0 

221,5 

51 

52 

716  003  343  634,8 

22  015  739  817,7 

225  774  434,3 

2  258  325,4 

22  583,3 

225,8 

52 

53 

724  275  869  600,8 

22  428  371  185,5 

230  115  100,3 

2  301  754,7 

23  017,6 

230,2 

53 

54 

732  393  759  823,0 

22  840  610  876,5 

234  455  722,9 

2  345  183,9 

23  451,9 

234,5 

54 

55 

740  362  689  494,2 

23  252  459  633,7 

238  796  302,1 

2  388  613,1 

23  886,2 

238,9 

55 

56 

748. 188  027  006,2 

23  663  918  197,8 

243  136  837,9 

2  432  042,3 

24  320,5 

243,2 

56 

57 

755  874  855  672,5 

24  074  987  307,4 

247  477  330,3 

2  475  471,5 

24  754,8 

247,5 

57 

58 

763  427  993  562,9 

24  485  667  699,2 

251  817  779,4 

2  518  900,7 

25  189,1 

251,9 

58 

59 
60 

770  852  Oil  642,1 

24  895  960  107,5 

256  158  185,1 

2  562  329,9 
2  605  759,1 

25  623,4 

256,2 
260,6 

59 
60 

778  151  250  383,6 

25  305  865  264,8 

260  498  547,4 

26  057,7 

61 

785  329  835  010,8 

25  715  383  901,3 

264  838  866,3 

2  649  188,3 

26  492,0 

264,9 

61 

62 

792  391  689  498,3 

26  124  516  745,5 

269  179  141,9 

2  692  617,4 

26  926,3 

269,3 

62 

63 

799  340  549  453,6 

26  533  264  523,3 

273  519  374.0 

2  736  046,6 

27  360,6 

273,6 

63 

64 

806  179  973  983,9 

26  941  627  959,0 

277  859  562,8 

2  779  475,8 

27  794,8 

277,9 

64 

65 

812  913  356  642,9 

27  349  607  774,8 

282  199  708,3 

2  822  905,0 

28  229,1 

282,3 

65 

66 

819  543  935  541,9 

27  757  204  690,6 

286  539  810,3 

2  866  334,1 

28  663,4 

286,6 

66 

67 

826  074  802  700,8 

28  164  419  424,5 

290  879  869,0 

2  909  763,3 

29  097,7 

291,0 

67 

68 

832  508  912  706,2 

28  571  252  692,5 

295  219  884,3 

2  953  192,4 

29  532,0 

295,3 

68 

69 

70 

838  849  090  737,3 

28  977  705  208,8 

299  559  856,2 

2  ^996  621,6 

29  906,3 

299,7 
304,0 

69 
70 

^45  098  040  014,3 

29  383  777  685,2 

303  899  784,8 

3  040  050,7 

30  400,6 

71 

851  258  348  719,1 

29  789  470  831,9 

308  239  670,0 

3  083  479,9 

30  834,9 

308,3 

71 

72 

857  332  496  431,3 

30  194  785  356,8 

312  579  511,8 

3  126  909,0 

31  269,2 

312,7 

72 

73 

863  322  860  120,5 

30  599  721  966,0 

316  919  310,3 

3  170  338,1 

31  703,5 

317,0 

73 

74 

869  231  719  731,0 

31  004  281  363,5 

321  259  065,4 

3  213  767,3 

32  137,8 

321,4 

74 

75 

875  061  263  391,7 

31  408  464  251,6 

325  598  777,1 

3  257  196,4 

32  572,1 

325,7 

75 

76 

880  813  592  280,8 

31  812  271  330,4 

329  938  445,5 

3  300  625,5 

33  006,4 

330,1 

76' 

77 

886  490  725  172,5 

32  215  703  298,0 

334  278  070,5 

3  344  054,6 

33  440,7 

334,4 

77 

78 

892  094  602  690,5 

32  618  760  850,7 

338  617  652,2 

3  387  483,7 

33  875,0 

338,7 

78 

79 
80 

897  627  091  290,4 

33  021  444  682,9 

342  957  190,4 

3  430  912,9 

34  309,3 

343,1 
347,4 

79 
80 

903  089  986  991,9 

33  423  755  486,9 

347  296  685,4 

3  474  342,0 

34  743,6 

81 

908  485  018  878,6 

33  825  693  953,3 

351  636  136,9 

3  517  771,1 

35  177,9 

351,8 

81 

82 

913  813  852  383,7 

34  227  260  770,6 

355  975  545,1 

3  561  200,2 

35  612,1 

356,1 

82 

83 

919  078  092  376,1 

34  628  456  625,3 

360  314  910,0 

3  604  629,2 

36  046,4 

360,5 

83 

84 

9^4  279  286  061,9 

35  029  282  202,4 

364  654  231,5 

3  648  058,3 

36  480,7 

364,8 

84 

85 

929  418  925  714,3 

35  429  738  184,5 

368  993  509,6 

3  691  487,4 

36  915,0 

369,2 

85 

86 

934  498  451  243,6 

35  829  825  252,8 

373  332  744,4 

3  734  916,5 

37  349,3 

373,5 

86 

87 

939  519  252  618,6 

36  229  544  086,3 

377  671  935,8 

3  778.345,6 

37  783,6 

377,8 

87 

88 

944  482  672  150,2 

36  628  895  362,2 

382  Oil  083,8 

3  821  774,6 

38  217,9 

382,2 

88 

89 
90 

949  390  006  644,9 

37  027  879  755,8 

386  350  188,6 

3  865  203,7 

38  652,2 

386,5 
390,9 

89 
90 

954  242  509  439,3 

37  426  497  940,6 

390  689  249,9 

3  908  632,7 

39  086,5 

91 

959  041  392  321,1 

37  824  750  588,3 

395  028  267,9 

3  952  061,8 

39  520,8 

395,2 

91 

92 

963  787  827  345,6 

38  222  638  368,7 

399  367  242,6 

3  995  490,9 

39  955,1 

399,6 

92 

93 

968  482  948  553,9 

38  620  161  949,7 

403  706  173,9 

4  038  919,9 

40  389,4 

403,9 

93 

94 

973  127  853  599,7 

39  017  321  997,4 

408  045  061,8 

4  082  348,9 

40  823,7 

408,2 

94 

95 

977  723  605  288,8 

39  414  119  176,1 

412  383  906,5 

4  125  778,0 

41  258,0 

412,6 

95 

96 

982  271  233  039,6 

39  810  554  148,4 

416  722  707,7 

4  169  207,0 

41  692,3 

416,9 

96 

97 

986  771  734  266,2 

40  206  627  574,7 

421  061  465,6 

4  212  636,0 

42  126,6 

421,3 

97 

98 

991  226  075  692,5 

40  602  340  114,1 

425  400  180,2 

4  256  065,1 

42  560,9 

425,6 

98 

99 

995  635  194  597,5 

40  997  692  423,5 

429  738  851,4 

4  299  494,1 

42  995,2 

430,0 

99 

32 


TABLE    OF   INTEREST    RATIOS 


1  +  i 

Logarithm 

1  +  i 

Logarithm 

1.00125 

1.0015 

1.00175 

1.002 

1.00225 

000  542  529  092  294 
000  650  953  629  595 
000  759  351  104  737 
000  867  721  531  227 
000  976  064  922  559 

1.01375 

1.014 

1.01425 

1.0145 

1.01475 

005  930  867  219  212 

006  037  954  997  317 
006  145  016  376  364 
006  252  051  369  365 
006  359  059  989  323 

1.0025 
1.00275 
1.003 
1.00325 
1 . 0035 

001  084  381  292  220 
001  192  670-653  684 
001  300  933  020  418 
001  409  168  405  876 
001  517  376  823  504 

1.015 

1.01525 

1.0155 

1.01575 

1.016 

006  466  042  249  232 
006  572  998  162  075 
006  679  927  740  826 
006  786  830  998  449 
006  893  707  947  900 

1.00375 
1.004 
1 . 00425 
1 . 0045 
1 . 00475 

001  625  558  286  737 
001  733  712  809  001 
001  841  840  403  709 

001  949  941  084  268 

002  058  014  864  072 

1.01625 

1.0165 

1.01675 

1.017 

1.01725 

007  000  558  602  125 
007  107  382  974  057 
007  214  181  076  625 
007  320  952  922  745 
007  427  698  525  323 

1.005 

1.00525 

1.0055 

1.00575 

1.006 

002  166  061  756  508 
002  274  081  774  949 
002  382  074  932  761 
002  490  041  243  299 
002  597  980  719  909 

1.0175 

1.01775 

1.018 

1.01825 

1.0185 

007  534  417  897  258 
007  641  111  051  437 
007  747  778  000  740 
007  854  418  758  035 
007  961  033  336  183 

1.00625 

1.0065 

1.00675 

1.007 

1.00725 

002  705  893  375  925 
002  813  779  224  673 

002  921  638  279  469 

003  029  470  553  618 
003  137  276  060  415 

1.01875 

1.019 

1.01925 

1.0195 

1.01975 

008  067  621  748  033  . 
008  174  184  006  426 
008  280  720  124  194 
008  387  230  114  159" 
008  493  713  989  132 

1.0075 
1.00775 
1.008 
1 . 00825 
1.0085 

003  245  054  813  147 
003  352  806  825  089 
003  460  532  109  506 
003  568  230  679  656 
003  675  902  548  784 

1.02 

1.02025 

1.0205 

1.02075 

1.021 

008  600  171  761  918 
008  706  603  445  309 
008  813  009  052  089 

008  919  388  595  035 

009  025  742  086  910 

1.00875 

1.009 

1.00925 

1.0095 

1.00975 

003  783  547  730  127 
003  891  166  236  911 

003  998  758  082  352 

004  106  323  279  658 
004  213  861  842  026 

1.02125 

1.0215 

1.02175 

1.022 

1.02225 

009  132  069  540  472 
009  238  370  968  466 
009  344  646  383  631 
009  450  895  798  694 
009  557  119  226  374 

1.01 

1.01025 

1.0105 

1.01075 

1.011 

004  321  373  782  643 
004  428  859  114  686 
004  536  317  851  323 
004  643  750  005  712 
004  751  155  591  001 

1.0225 

1.02275 

1.023 

1.02325 

1.0235 

009  663  316  679  379 
009  769  488  170  411 
009  875  633  712  160  ' 

009  981  753  317  307 

010  087  846  998  524 

1.01125 

1.0115 

1.01175 

1.012 

1.01225 

004  858  534  620  329 

004  965  887  106  823 

005  073  213  063  604 
005  180  512  503  780 
005  287  785  440  451 

1.02375 

1.024 

1.02425 

1.0245 

1.02475 

010  193  914  768  475 
010  299  956  639  812 
010  405  972  625  180 
010  511  962  737  214 
010  617  926  988  539 

1.0125 

1.01275 

1.013 

1.01325 

1.0135 

005  395  031  886  706 
005  502  251  855  626 
005  609  445  360  280 
005  716  612  413  731 
005  823  753  029  028 

1.025 
1 . 02525 
1.0255 
1.02575 
1.026 

010  723  865  391  773 
010  829  777  959  522 
010  «35  664  704  385  • 
Oil  041  525  638  950 
Oil  147  360  775  797 

TABLE    OF   INTEREST    RATIOS— 
Continued 

TABLE    OF   SUB-RECIPROCALS 

(Art.  51)                             33 

1  +  i 

Logarithm 

Initial  Figures 

Sub-reciprocal 

1.02625 

1.0265 

1.02675 

1.027 

1.02725 

Oil  253  170  127  497 
Oil  358  953  706  611 
on  464  711  525  690 
Oil  570  443  597  278 
Oil  676  149  933  909 

10 
11 
12 
13 
14 

90 
83 
76 
71 

66 

1.0275 

1.02775 

1.028 

1.02825 

1.0285 

Oil  781  830  548  107 
Oil  887  485  452  387 
Oil  993  114  659  257 
012  098  718  181  213 
012  204  296  030  743 

15 

16    ' 
17 
18 
19 

62 
58 
55 
52 
50 

1 . 02875 
1.029 
1 . 02925 
1.0295 
1.02975 

012  309  848  220  326 
012  415  374  762  433 
012  520  875  669  524 
012  626  350  954  050 
012  731  800  628  455 

20 
21 
22 
23 
24 

47 
45 
43 
41 
40 

1.03 

1.0305 

1.031 

1.0315 

1.032 

012  837  224  705  172 

013  047  996   115  232 
013  258  665  283  517 
013  469  232  309   170 

•   013  679  697  291   193 

25 
26 

27 
28 
29 

38 
37 
35 
34 
33 

1.0325 
1.033 
1 . 0335 
1.034 
1.0345 

013  890  060  328  439 

014  100  321   519  621 
014  310  480  963  307 
014  520  538  757  924 
014  730  495  001  753 

30 
31 
32 
33 
34 

32 
31 
30 
29 

28 

1.035 
1 . 0355 
1.036 
1.0375 
1.038 

014  940  349  792  937 

015  150  103  229  471 
015  359  755  409  214 

015  988  105  384  130 

016  197  353  512  439 

35-36 

37 
38-39 

40 
41-42 

27 
26 
25 
24 
23 

1.039 
1.04 
1.041 
1 . 0425 
1 .  043 

016  615  547  557  177 

017  033  339  298  780 

017  450  729  510  536 

018  076  063  645  795 
018  284  308  426  531 

43-44 
45-46 
47-49 
50-51 
52-54 

22 
21 
20 
19 
18 

1.044 

1.045 

1.046 

1.0475 

1.048 

018  700  498  666  243 

019  116  290  447  073 

019  531  684  531  255 

020  154  031  638  333 
020  361  282  647  708 

55-57 
58-61 
62-65 
66-70 
71-75 

17 
16 
15 
14 
13 

1.049 

1.05 

1.055 

1.06 

1.065 

020  775  488  193  558 

021  189  299  069  938 
023  252  459  633  711 
025  305  865  264  770 
027  349  607  774  757 

76-82 

83-89 

90 

12 

11 

1 

1.07 

1.075 

1.08 

1.09 

1.10 

029  383  777  685  210 
031  408  464  251  624 
033  423  755  486  950 
037  426  497  940  624 
041  392  685  158  225 

* 

•• 

34 


TABLE    OF    MULTIPLES 


1 

2 

3 

4 

5 

6 

7 

8 

9 

001 

002 

003 

004 

005 

006 

007 

008 

009 

002 

004 

006 

008 

010 

012 

014 

016 

018 

003 

006 

009 

012 

015 

018 

021 

024 

027 

004 

008 

012 

016 

020 

024 

028 

032 

036 

005 

010 

015 

020 

025 

030 

035 

040 

045 

006 

012 

018 

024 

030 

036 

042 

048 

054 

007 

014 

021 

028 

035 

042 

049 

056 

003 

008 

016 

024 

032 

040 

048 

056 

064 

072 

009 

018 

027 

036 

045 

054 

063 

072 

081 

010 

020 

030 

040 

050 

060 

070 

080 

090 

Oil 

022 

033 

044 

055 

066 

077 

088 

099 

012 

024 

036 

048 

060 

072 

084 

096 

108 

013 

026 

039 

052 

065 

078 

091 

104 

117 

014 

028 

042 

056 

070 

084 

098 

112 

126 

015 

030 

045 

060 

075 

090 

105 

120 

135 

016 

032 

048 

064 

080 

096 

112 

128 

144 

017 

034 

051 

068 

085 

102 

119 

136 

153 

018 

036 

054 

072 

090 

108 

126 

144 

162 

019 

038 

057 

076 

095 

114 

133 

152 

171 

020 

040 

060 

080 

100 

120 

140 

160 

180 

021 

042 

063 

084 

105 

126 

147 

168 

189 

022 

044 

066* 

088 

110 

132 

154 

176 

198 

023 

046 

069 

092 

115 

138 

161 

184 

207 

024 

048 

072 

096 

120 

144 

168 

192 

210 

025 

050 

075 

100 

125 

150 

175 

200 

225 

026 

052 

078 

104 

130 

156 

182 

208 

234 

027 

054 

081 

108 

135 

162 

189 

216 

243 

028 

056 

084 

112 

140 

168 

196 

224 

252 

029 

058 

087 

116 

145 

174 

203 

232 

261 

030 

060 

090 

120 

150 

180 

210 

240 

270 

031 

062 

093 

124 

155 

186 

217 

248 

279 

032 

064 

096 

128 

160 

192 

224 

256 

288 

033 

066 

099 

132 

165 

198 

231 

264 

297 

034 

068 

102 

136 

170 

204 

238 

272 

306 

035 

070 

105 

140 

175 

210 

245 

280 

315 

036 

072 

108 

■  144 

180 

216 

252 

288 

324 

037 

074 

111 

148 

185 

222 

259 

296 

333 

038 

076 

114 

152 

190 

228 

266 

304 

342 

039 

078 

117 

156 

195 

234 

273 

312 

351 

040 

080 

120 

160 

200 

240 

280 

320 

360 

041 

082 

123 

164 

205 

246 

287 

328 

369 

042 

084 

126 

168 

210 

252 

* 

294 

336 

378 

043 

086 

129 

172 

215 

258 

301 

344 

387 

044 

088 

132 

176 

220 

264 

308 

352 

396 

045 

090 

135 

180 

225 

270 

315 

360 

405 

046 

092 

138 

184 

230 

276 

322 

368 

414 

047 

094 

141 

188 

235 

282 

329 

376 

423 

048 

096 

144 

192 

240 

288 

336 

384 

432 

049 

098 

147 

196 

245 

294 

343 

392 

441 

TABLE    OF    MULTIPLES— Continued 


35 


1 

2 

3 

4 

5 

6 

7 

8 

9 

050 

100 

150 

200 

250 

300 

350 

400 

450 

051 

102 

153 

204 

255 

306 

357 

408 

459 

052 

104 

156 

208 

260 

312 

364 

416 

468 

053 

106 

150 

212 

265 

318 

371 

424 

477 

054 

108 

162 

216 

270 

324 

378 

432 

486 

055 

110 

165 

220 

275 

330 

385 

440 

495 

056 

112 

168 

224 

280 

336 

392 

448 

504 

057 

114 

171 

228 

285 

342 

399 

456 

513 

058 

116 

174 

232 

290 

348 

406 

464 

522  1 

059 

118 

177 

236 

295 

354 

413 

472 

531 

060 

120 

180 

240 

300 

360 

420 

480 

540 

061 

122 

183 

244 

305 

306 

427 

488 

549 

002 

124 

186 

248 

310 

372 

434 

496 

558 

063 

126 

189 

252 

315 

378 

441 

504 

567  i 

064 

128 

192 

256 

320 

384 

448 

512 

576 

065 

130 

195 

260 

325 

390 

455 

^20 

585 

066 

132 

198 

264 

330 

396 

462 

528 

594 

067 

134 

201 

268 

335 

402 

469 

536 

603  i 

068 

136 

204 

272 

340 

408 

476 

544 

612  •! 

069 

138 

207 

276 

345 

414 

483 

552 

621  ' 

070 

140 

210 

280 

350 

420 

490 

560 

630 

071 

142 

213 

284 

355 

426" 

497 

568 

639  1 

072 

144 

216 

288 

360 

432 

504^ 

576 

648  1 

073 

146 

219 

292 

365 

438 

511 

584 

657 

074 

148 

222 

296 

370 

444 

518 

592 

666 

075 

150 

225 

300 

375 

450 

525 

600 

675 

076 

152 

228 

304 

380 

456 

532 

608 

684 

077 

154 

231 

308 

385 

462 

539 

616 

693 

078 

156 

234 

312 

390 

468 

546 

624 

702 

079 

158 

237 

316 

395 

474 

553 

632 

711 

080 

160 

240 

320 

400 

480 

500 

640 

720 

081 

162 

243 

324 

405 

486 

567 

048 

729 

082 

164 

246 

328 

410 

492 

574 

656 

738 

083 

166 

249 

332 

415 

498 

581 

664 

747 

084 

168 

252 

336 

420 

504 

588 

672 

756 

085 

170 

255 

340 

425 

510 

595 

680 

765 

086 

172 

258 

344 

430 

516 

602 

688 

774 

087 

174 

261 

348 

435 

522 

609 

696 

783 

088 

176 

264 

352 

440 

528 

616 

704 

792 

089 

178 

267 

356 

445 

534 

623 

712 

801 

090 

180 

270 

360 

450 

540 

630 

720 

810 

091 

182 

273 

364 

455 

546 

637 

728 

819 

092 

184 

276 

368 

460 

552 

644 

736 

828 

093 

186 

279 

372 

465 

558 

651 

744 

837 

094 

188 

282 

376 

470 

564 

658 

752 

846 

095 

190 

285 

380 

475 

570 

665 

760 

855 

096 

192 

288 

384 

480 

576 

672 

768 

864 

097 

194 

291 

388 

485 

582 

679 

776 

873 

098 

196 

294 

392 

490 

588 

686 

784 

882 

099 

198 

297 

396 

495 

594 

693 

792 

891 

36 


SPECIMEN  OF  RULED  PAPER 

RECOMMENDED  FOR  USE  WITH  THE  FOREGOING  TABIvES. 


1 

1      1 

i      1 

.      i 

i   1^ 

!      1 

1 

■ 

j.    i 

L 

- 

— 

— 

— 

__ 

! 

1 

. 

' 

! 

1 

i 

i 

T 

i 

! 
1— 

i 

1      i 

i 

r       •: 

i 

j 

i 

j 

I 

T  - 

j 

' 

1       ! 

i 

7"!  ■ 
1     ■ 

1 
t 

!      1 

1 

1      : 
i      i 
i      i 

1 

1 

.,J_„_ 

I 
! 

- 

i 

i 

.-,_,..  j.._.,^-_ 

■  r  -! 

PART  III. 


The  Doctrin  of  Interest 


PART  III. 


THE  DOCTRIN  OF  INTEREST. 


Interest. 


67.  — Interest,  mathematically  considerd,  is  the  increase 
of  an  indettedness  by  lapse  of  time.  The  rate  of  such  increase 
varies  with  circumstances,  *  and  is  subject  to  bargaining  ;  the 
resulting  contract,  exprest  or  implied,  must  embody  the  fol- 
lowing terms: 

Principal.  The  number  of  units  of  value  (dollars,  pounds, 
francs,  marks,  etc.,)  originally  loand  or  invested. 

Interest  Rate.  The  fraction  which  is  added  to  each 
unit  by  the  lapse  of  one  unit  of  time ;  usually  a  small  decimal. 

Frequency.  The  length  of  the  unit  of  time,  measured 
in  years,  months  or  days. 

Time.  The  number  of  units  of  time  during  which  the 
indettedness  is  to  continue. 

68. — As  each  dollar  increases  just  as  much  as  every  other 
dollar,  it  is  best  at  first  to  consider  the  principal  as  one  dollar 
and  when  the  proper  function  thereof  has  been  calculated, 
to  multiply  it  by  the  number  of  dollars. 

69. — The  interest  rate  is  usually  spoken  of  as  so  much 
percent  per  period  or  term.  "6%'  per  annum"  means  an 
increase  of  .  06  for  each  term  of  a  year.  We  will  designate 
the  interest  rate  by  the  letter  i ;  as,  ?  =  .06.  At  the  end  of 
one  term  the  increast  indettedness  is  1  -f  /,  (1.06),  a  very 
important  quantity  in  computation. 


*  For  discussion  of  the  causes  for  higher  or  lower  interest  rates,  see 
The  Rate  of  Interest,  by  Prof.  Irving  Fisher. 


40  The  Doctrin  op  Interest. 

70. — Punctual  Interest.  The  usual  contract  is  that  the 
increase  shall  be  paid  off  in  cash  at  the  end  of  each  period, 
restoring  the  principal  to  its  original  quantity.  Let  c  denote 
the  cash  payment ;  then  1  +  z  —  ^r  =  1 ;  and  the  second  term 
would  repeat  the  same  process.  The  payment  of  cash  for 
interest  must  not  be  regarded  as  the  interest :  it  is  a  cancel- 
lation of  part  of  the  increast  principal.  Many  persons,  and 
even  courts,  have  been  misled  by  the  old  definition  of  interest, 
"money  paid  for  the  use  of  money,"  into  treating  uncollected 
or  unmatured  interest  as  a  nullity,  tho  secured  precisely  in  the 
same  way  as  the  principal. 

71. — But  the  interest  money  may  not  be  paid  exactly  at 
the  end  of  each  term,  either  in  violation  of  the  contract  or  by 
a  special  clause  permitting  it  to  run  on,  or  by  the  det  being 
assigned  to  a  third  party  at  a  price  which  modifies  the  true 
interest  rate.  In  this  case  the  question  arises  :  how  shall  the 
interest  be  computed  for  the  following  periods  ?  This  gives 
rise  to  a  distinction  between  simple  and  compound  interest. 

72. — Simple  Interest.  During  the  second  period,  altho 
the  borrower  has  in  his  hands  an  increast  principal,  1  +  i,  he 
is  at  simple  interest  only  charged  with  interest  on  1,  and  has 
the  free  use  of  /,  which  tho  small  has  an  earning  power  pro- 
portionate to  that  of  1.  His  indettedness  at  the  end  of  the 
second  term  is  1  +  2/,  and  thereafter  1  +  3/,  1  -f-  4/,  etc. 
After  the  first  period  he  is  not  charged  with  the  agreed  per- 
centage of  the  sum  actually  employed  by  him,  and  this  to  the 
detriment  of  the  creditor.  For  any  scientific  calculation, 
simple  interest  is  impossible  of  application. 

73.— Compound  Interest.  The  indettedness  at  the  end 
of  the  first  period  is  1  H-  /,  and  up  to  this  point  punctual,  simple 
and  compound  interest  coincide.  But  in  compound  interest  the 
fact  is  recognized  that  the  increast  principal,  1  -f  z,  is  all  sub- 
ject to  interest  during  the  next  period,  and  that  the  det 
increases  by  geometrical  progression,  not  arithmetical.  The 
increase  from  1  to  1  +  z  is  regarded,  not  as  an  addition  of  i  to 
1,  but  as  a  multiplication  of  1  by  the  ratio  of  increase 
(1  -f  z) .  We  shall  designate  the  ratio  of  increase  by  r  when 
convenient,  altho  this  is  merely  an  abbreviation  of  1  H-  z,  and 
the  two  expressions  are  at  all  times  interchangeable. 


Thk  Amount.  41 

74. — For  the  second  period,  1  +  /  is  the  actual  and 
equitable  principal,  and  it  should  be  again  increast  in  the  ratio 
1  -j-  i.  The  total  indettedness  at  the  end  of  the  second  period 
is  therefore  1  x  (1  +  0  x  (1  +  z)  =  (1  +  0'  =  r\  At 
the  end  of  the  third  period  it  will  have  become  r^,  and  at  the 
end  of  term  No.  /,  r^. 

Thb  Amount. 

75. — The  sum  to  which  $1  will  have  increast  at  compound 
interest  at  ^  (or  100/  per  cent.)  in  /  periods,  is  called  the 
Amount,  and  will  be  designated  as  s.  We  then  have  the 
following  equation: 

5  =  rt=  (1  +  iy 

76. — To  find  the  amount  of  one  dollar,  raise  the 
ratio  to  a  power  whose  exponent  is  the  number  of 
periods. 

77. — The  logarithm  of  the  ratio  of  increase  is  the  most 
important  logarithm  for  interest  calculations.  If  the  interest 
rate  does  not  exceed  two  figures,  the  logarithm  will  be  found 
in  full  in  col.  B,  TF.  For  convenience  we  will  designate  it  by 
a  capital  letter  L.  Thus,  if  i  =  .065,  L  will  be  found  opposit 
Q^  in  B.     If  i  =  .065  ;  log.  r  =  I,  =  .027  349  607  774,8. 

78. — As  powers  are  found  by  multiplying  the  logarithm, 
L  must  be  multiplied  by  i. 

r^  nl         tl, 

79. — To  find  the  amount,  multiply  the  logarithm  of 
the  ratio  by  the  number  of  periods,  and  the  correspond- 
ing number  will  be  the  amount  of  $1. 

80. — Let  the  interest  rate  be  3.5%' per  annum,  payable 
annually,  what  will  be  the  amount  of  $1  at  the  end  of  100 
years?  Turning  to  col.  B,  TF,  we  find  opposit  B  35,  (or 
1.035)  the  logarithm  .014  940  349  792,9. 

1.=    .014  940  349  792,9 

/  =  100  /L  =  1 .  494  034  979  29 

From  the  characteristic  1,  it  appears  that  the  amount  will  be 
in  the  tens  of  dollars  ;  and  as  the  decimal  part  of  the  logarithm 
is  a  little  more  than  that  which  is  opposit  31  we  know  that  the 
amount  is  $31  and  some  cents.  Thus  a  rough  idea  of  the 
amount  may  be  gaind  almost  instantly. 


42  The  Doctrin  of  Interest. 

81. — To  obtain  a  more  accurate  value  and  one  which  will 
be  sufi&ciently  near  for  a  large  principal,  we  proceed  as  follows: 

82. — In  the  first  place  we  can  only  obtain  ten  correct 
figures  from  lOOL.  The  final  figure  9  is  never  perfect ;  it  may 
be  8 .  51  or  9 .  49  or  anywhere  between.  We  must,  therefore,  use 
only  eleven  in  the  logarithm  and  finally  get  ten  in  the  number. 

(?)  nl       494  034  979  29 


A  31 

491  361  693  83 

2  673  285  46 

B  06 

2  597  980  72 

75  304  74 

C  17 

73  823  79 

1  480  95 

D  34 

1  476  60 

4  35 

E  10 

4  34 

F    2 

1 

31 

B  06 

186 

31186  .... 

C     1 

31186 

7 

2  18  3  0  2 

3119130162 

D    3 

9357 

4 

1248 

3119140767 

E  10 

3  1 

3  119  140  798  =  5  (/ 100,  z  .03) 

83. — In  order  to  give  accurate  results  up  to  twelve  figures 
for  one  hundred  interest  terms,  we  have  provided  on  page  32 
a  special  table  of  the  logarithms  of  the  150  interest  ratios  (1  +  i) 
which  most  frequently  occur,  calculated  to  15  places,  which 
allows  two  places  for  loss  in  multiplication. 


43 
The  Present  Worth. 

84. — The  sum  which  if  now  invested  at  i  will  in  /  periods 
amount  to  $1  is  evidently  less  than  $1.  It  is  in  the  same  pro- 
portion to  1  as  1  is  to  s.  Designating  the  present  worth  by  /, 
we  have 

p  \\  \\\\  s 

ox  p  =—   =  5~^ 
s 

or  the  amount  and  the  present  worth  are  reciprocals  of  each 

other. 

A  series  of  amounts  reads 

1,  r\  r%  r%  r*,  r",  etc. 
A  series  of  present  worths  reads 

1,  r"\  r~%  r  %  r"*,  r~%  etc. 
Reversing  the  latter  series  and  connecting  it  with  the 
former  we  have  a  continuous  series  in  geometrical  progression: 
r"%  r"*,  r~^,  r"%  r"\  1,  r\  r^  r^  r"*,  r^ . 
Using  1.03  as  the  ratio,  the  series  becomes 

f'  .86260878 

r-^  .88848705 

r-«  .91514166 

r"'  .94259591 

r'  .97087379 

r°  1. 

r*  1.03 

r'  1.0609 

r^  1.092727 

r*  1.12550881 

r"^  1.15927407 

In  this  series,  which  might  be  extended  indefinitly  upward 
and  downward,  every  term  is  a  present  worth  of  any  which 
follows  it  and  an  amount  of  each  which  precedes  it.  .86260878 
is  the  present  worth  at  10  interest  periods  of  1 .  15927407  ; 
1.12550881  is  the  amount  at  eight  periods  of  .88848705. 

85.  — If  any  term  be  multiplied  by  1.03,  the  product  will 
be  the  next  following  term  ;  if  it  be  divided  by  1.03  or  (which 
is  the  same  thing)  be  multiplied  by  .97087379,  the  product 
will  be  the  next  preceding  term. 


44  I^HB  DocTRiN  OF  Interest. 

86. — To  find  the  logarithm  of  the  present  worth,  subtract 
the  logarithm  of  the  amount  (for  the  same  time)  from  zero. 

In  the  preceding  example,  but  using  L  from  the  15  place 
table 

s        nl       /fL  =  1.494  034  979  293,7 

p=^\ls    nl   —/L  =  2.505  965  020  706,3 

A  32  505  149  978  319,9 

815  042  386,4 

B  01  434  077  479,3 

380,964  907,1 

C   87  377  671935.8 

3  292  971,3 

D  75  3  257  196,4 

35  774,9 

K  82  35  612.1 

162,8 
F  37  160,7 

G  49  2,1 


32 
3  01  032 

3  2  032  .  .  .  . 
C    8  256256 

7  224224 

32  0-5  986784.  .  . 
D  7  22441907 

5       1602993 

3206010828900 

E  8  256481 

2  6412 

F  3  962 

7  22 

G  4  2 

9      

3206011092779 
,0  32060110928=;^  (1.035)^°°  to  12  places. 


The  Compound  Interest  and  Discount.  45 

87. — That  the  amount  and  the  present  worth  are  correct 
reciprocals  may  be  tested  by  multiplying  them  together. 
Taking  a  few  figures  of  each  we  have 

31  ■  19  14 
.03 


2 

06 
01 


935742 

62383 

1871 

1 


1  .  00000 
Every  pair  of  reciprocals  gives  a  product  of  1. 


The  Compound  Interest  and  Discount. 

88. — We  have  hitherto  used  the  word  "interest" 
abstractly  as  denoting  that  force  or  principle  which  effects  the 
increase  of  the  amount  of  an  indettedness  as  time  goes  on.  The 
interest-increment  which  is  thus  added  is  also  frequently  called 
"  the  Interest,"  which  may  be  v/ritten  with  a  capital  letter. 

89. — If  we  take  the  original  principal  away  from  the 
amount,  we  evidently  have  the  Interest.     For  a  single  period 

2  =  1  +  z  _  1  =  r  —  1. 

When  there  are  more  than  one  period  it  is  the  compound 
Interest,  obtaind  in  the  same  way  and  represented  by  a 
capital  I  =  (1  +  z)^  —  1  =  ?-*  —  1  ==  .S  —  1. 

Thus  the  compound  Interest  of  $1  at  3%  per  period  for 
100  periods  is  $31.19  —  1.00  =  $30.19.  For  two  periods  it 
is  1.0609  —  1  =  0.0609. 

90. — In  the  opposit  case  of  a  present  worth  there  is  a 
diminution  of  the  principal.  The  present  worth  of  $1  at  3  per 
cent,  one  period,  is  .97087379;  the  Discount  is  not  .03,  but 
.02912621,  the  true  principal  being  not  $1,  but  .97087379, 
which  X  .  03  =  .  02912621.  Representing  the  simple  Discount 
by  d,  we  have  d  —\  —  p  =  i  y,  p  =  i/s. 

91. — If  there  be  more  than  one  term  involvd,  it  is  com- 
pound Discount,  which  will  be  represented  by  D.  Thus,  at 
3  per  cent  for  5  periods  D  =  1  —  .86261  =  .13739.  D  is 
also  the  present  worth  of  the  compound  Interest  for  the  same 
time.     .15927  x  .86261  =  .13739. 


46  The  Doctrin  of  Interest. 

In  general  D  =  l  —  p  —  \p  =  1/s, 

92.— Thus  we  see  that  the  variance  from  par  (|1)  is 
called  compound  Interest  or  compound  Discount,  according 
as  regarded  from  the  past  or  the  future  point  of  view  and  that 
their  properties  are  as  follows: 

D  ^  1  —  ^t 
and  their  relation  is  D  =  /I ;  or  I  =  ^D. 


Finding  Time  or  Rate. 

93.— 'By  time  we  mean  the  number  of  periods,  terms  or 
intervals,  and  by  this  number  the  logarithm  of  the  interest- 
ratio  is  multiplied  to  produce  the  logarithm  of  the  amount. 

(/  X  L)  /«  5 
/  X  log  (1  +  i)  =  log  s 

94. — If  the  amount  is  known  and  the  rate,  but  the  num- 
ber of  periods  unknown,  we  can  transform  the  above  equation 
into  this: 

log  s 

h 

95. — At  .03  interest,  in  how  many  periods  will  $1  amount 
to  $2,  or  how  long  will  it  take  a  sum  to  double  itself  ? 
logs  =  log2        =  .3010299956640 
I,  =  log  1.03=  .0128372247052 

Using  only  seven  places 

.  0128372)  .  3010300  (23.47 

2  5  6  7  4  4 

442860 

385116 

57744 

49349 

8395 

The  money  will  double  in  24  periods,  as  it  is  not  quite 
doubled  at  23. 

96.' — How  many  periods  must  a  det  of  $1  be  deferd  to  be 
worth  now  30  cents,  at  3>^%  ? 


Finding  Time  or  Rate.  47 

Lo^  1.035  =  .01494035 
Log  1.035-'  =  1.98505965 
Log-    .30      =  1.47712125 

98  5  05965)    1.47712125( 


—  .01494035)— .52287875(34.9997 

4  4  8  2  10  5 
7466825 
5976140 
1490685 
1344631 
146054 
134463 
11591 
Practically  35  periods. 
For  convenience  in  division,  the  minus  sign  is  made  to 
extend  over  the  entire  logarithms.     Then,  as  both  divisor  and 
dividend  are  of  the  same  sign,  the  quotient  is  positiv. 

97. — If  the  rate  be  unknown,  the  equation/  x  log.  (1  -|-  z) 
=  log.   s  may  again  be  transformed  to 

log.(l  +  0='-^. 

98. — 20  periods  having  elapst  and  the  amount  of  $1  being 
now  $3.20713547,  what  is  the  rate? 

log.  3.20713547  =  .506117303 
.506117303/20  =  .025305865  In  1.06 
1  +  1  =  1.06.-.  z=  .06 


The  Annuity. 
99. — We  have  now  investigated  the  two  fundamental 
problems  in  compound  interest :  viz. ,  to  find  the  amount  of  a 
present  worth,  and  to  find  the  present  worth  of  an  amount. 
The  next  question  is  a  more  complex  one  :  to  find  the  amount 
and  the  present  worth  of  a  series  of  payments.  If  these  pay- 
ments are  irregular  as  to  time,  amount  and  rate  of  interest,  the 
only  way  is  to  make  as  many  separate  computations  as  there 
are  sums  and  then  add  them  together.  But  if  the  sums,  times 
and  rate  are  uniform,  we  can  devise  a  method  for  finding  the 
amount  or  present  worth  at  one  operation. 


48  The  Doctrin  of  Interest. 

100. — Annuity.  A  series  of  payments  of  like  amount, 
made  at  regular  periods,  is  called  an  annuity,  even  though  the 
period  be  not  annual,  but  a  half  year,  a  quarter  or  any  other 
length  of  time.  Thus,  if  an  agreement  is  made  for  the  follow- 
ing payments: 

On  Sept.   9  1904  |5100. 

On  March  9  1905  100. 

On  Sept.  9  1905  100. 

and  on  March  9  1906  100. 

this  would  be  an  annuity  of  $100  per  period,  terminating  after 
4  periods.  It  is  required  to  find  on  March  9,  1904,  assuming 
the  rate  of  interest  as  3%  per  period  :  First,  what  will  be  the 
total  amount  to  which  the  annuity  will  have  accumulated  on 
March  9,  1906  ;  second,  what  is  now,  on  March  9,  1904,  the 
present  worth  of  this  series  of  future  sums  ?  It  is  evident  that 
the  answer  to  the  first  question  will  be  greater  than  ;^400,  and 
that  the  answer  to  the  second  question  will  be  less  than  $400. 


Amount  of  an  Annuity. 

101. — It  is  easy,  in  this  case,  to  find  the  separate  amounts 
of  the  payments,  for  the  number  of  terms  is  very  small,  and 
we  have  already  computed  the  corresponding  values  of  $1.00. 
The  last  $100  will  have  no  accumulation,  and  will 

be  merely $100. 

The  third  $100  will  have  earned  in  one  period  |3.00, 

and  will  amount  to 103 . 

The  second  $100  will  amount  to 106.09 

The  first  $100  (rounded  off  at  cents)  will  amount  to      109 .  27 

and  the  total  amount  will  be $418.36 

102. — If,  however,  there  were  50  terms  instead  of  4,  the 
work  of  computing  these  50  separate  amounts,  even  by  the  use 
of  logarithms,  would  be  very  tedious. 

103. — Let  us  write  down  the  successiv  amounts  of  $1.00 

under  one  another: 

a 

Amounts  of  $1. 
1.00 
1.03 
1.0609 
1.092727 


Amoun'T  of  an  Annuity.  49 

104. — Now,  as  we  have  the  right  to  take  any  principal  we 
choose  and  multiply  it  by  the  number  indicating  the  value  of 
$1.00,  let  us  assume  one  dollar  and  three  cents,  and  multiply 
each  of  the  above  figures  by  1.03,  setting  the  products  in  a 
second  colum: 

a.  b.  c. 

Amounts  of  %\ .  00  Amounts  of  %\ .  03  Amounts  of  |0 .  03 

1.00  1.03 

1.03  1.0609 

1.0609  1.092727 

1.092727  1.12550881 

105. — Our  object  in  doing  this  was  by  subtracting  colum 
a  from  b  to  find  the  amount  of  an  annuity  of  three  cents. 
Before  subtracting,  we  have  the  right  to  throw  out  any  num- 
bers which  are  identical  in  the  two  colums.  Expunging  these 
like  quantities,  we  have  left  only  the  following: 

a.  b.                                 c. 

Annuity  of  $1 .  00  Annuity  of  $1 .  03             Annuity  of  |0 .  03 

1.00  1.12550881 

less  1.00 


1.12550881  Amount  0.12550881 

That  is,  an  annuity  of  three  cents  will  amount,  under  the 
conditions  assumed,  to  twelve  cents  and  the  decimal  550881. 
Therefore,  an  annuity  of  one  cent  will  amount  to  one-third  of 
.12550881  or  .04183627.  An  annuity  of  ^1.00  will  amount 
to  100  times  as  much,  or  ;^4. 183627,  which  agrees  exactly 
with  the  result  obtained  by  addition,  in  Article  45. 

106.— The  number  .12550881  (  obtaind  by  subtracting 
1 .00  from  1 .  12550881)  is  actually  the  compound  Interest  for  the 
given  rate  and  time,  and  the  number  .03  is  the  single  Interest) 
the  amount  of  the  annuity  of  ^1.00  is  .12550881  -f-  .03  = 
4.183627.  This  suggests  another  way  of  looking  at  it.  The 
compound  Interest  up  to  any  time  is  really  the  amount  of  a 
smaller  annuity,  one  of  three  cents  instead  of  a  dollar,  con- 
structed on  exactly  the  same  plan,  and  used  as  a  model. 

107.— Rule.  To  find  the  amount  of  an  annuity  of  ^1.00 
for  a  given  time  and  rate,  divide  the  compound  Interest  by  a 
single  Interest,  both  exprest  decimally . 


50  The  Doctrin  of  Interest. 

108.  — Let  S  and  P  represent  the  amount  and  the  present 
worth,  not  of  a  single  $1.00,  but  of  an  annuity  of  $1,  then 
S  =  I  -f-  /. 

Exprest  in  symbols  the  reasoning  would  be  this: 

Amount  of  annuity  of  1  =  r*-^  -f r^-{-  r^  +  r  +  1       (a) 

Multiplying  by  r. 

Amount  of  annuity  oi  1 -\-  i  =  r^  -\-  r^'^ .  .r*  +  r^-\-r^  ■{•  r         {b) 

Subtracting  (a)  r^-'^ r^  +  ^M-  r  +  1 

Amount  of  annuity  of  /    —  r^        0      0      0      0      0  —  1  (^:) 

=  r^  — 1  =  I 
Amount  of  annuity  of  1   =  S  =  1/i 

109. — If  the  number  of  periods  were  50,  insted  of  4,  the 
advantage  of  this  process,  with  the  use  of  logarithms,  will  be 
very  evident. 

The  rate  being  .03,  the  logarithm  of  the  ratio,  or 
L  =  .012  837  224  705  172 
50L  ==  .641861235  258,6 
Factors,  A  43  B  19  C  50  D  34  E  60  F  10  G  14 
5=        4  .  38390601876 

-  1  

I =        3  .  38390601876 

I->. 03  =  112.  796867292  =  8 
Compare  this  with  the  diflSculty  of  finding  the  result  by 
arithmetic  for  even  ten  periods. 


Present  Worth  of  an  Annuity. 
110. — To  find  the  present  worth  of  an  annuity,  we  can,  of 
course,  find  the  present  worth  of  each  payment  and  add  them 
together  ;  but  it  will  evidently  save  a  great  deal  of  labor  if  we 
can  derive  the  present  worth  immediately,  as  we  have  learnd 
to  do  with  the  amount. 

111. — The  like  course  of  reasoning  will  give  us  the  result. 
Take  the  four  numbers  representing  the  present  worths  of  $1.00 
at  4,  3,  2  and  1  periods  respectivly,  and  multiply  each  by  1.03. 
a.  b. 

Present  Worth  of  Present  Worth  of 

Annuity  of  $1 .  00  Annuity  of  $\  .  03 

.888487  .915142       • 

.915142  .942596 

.942596  .970874 

.970874  1.000000 


Present  Worth  of  an  Annuity.  51 

Canceling  all  equivalents,  we  have  c, 

.  888487  Present  Worth  of 

Annuity  of  .03 

1.000000 

1.000000  less  .888487 

.111513 
Annuity  of  $1 .  00  =  .  111513  -H  .  03  =     3 .  71710 
This  is  the  same  result   (rounded   up)  as    that  obtaind  by 
adding  column  a. 

112.— But  .111513  is  the  compound  discount  of  $100  for 
four  periods,  and  we  therefore  construct  this  rule: 

113. — Rule.  To  find  the  present  worth  of  an  annuity  of 
$1.00  for  a  given  time  and  rate,  divide  the  compound  Discount 
for  that  time  and  rate  by  a  single  interest.  Symbolically 
P  =  D  -f-  /.  We  might  give  this  the  form  P  =  S  -^  5,  being 
the  present  worth  of  the  amount  of  the  annuity. 

114. — It  may  assist  in  acquiring  a  clear  idea  of  the  work- 
ing of  an  annuity,  if  we  analyse  a  series  of  annuity  payments 
from  the  point  of  view  of  the  purchaser. 

115.— He  who  invests  $3.7171  at  3%,  in  an  annuity  of  4 
periods,  expects  to  receive  at  each  payment,  besides  3%  on  his 
principal  to  date,  a  portion  of  that  principal,  and  thus  to  have 

his  entire  principal  gradually  repaid. 

Principal. 

His  original  principal  is 3 .  7171 

At  the  end  of  the  first  period  he  receives  1 .00,  con- 
sisting of  3%  on  3.7171. .1115 

and  payment  on  principal .8885 .8885 

leaving  new  principal 2 .  8286 

(or  present  worth  at  3  periods). 

In  the  next  instalment 1  00 

there  is  interest  on  2.8286 .0849 

and  payment  on  principal 9151  .9151 

leaving  new  principal 1 .9135 

Third  instalment 1 .00 

Interest .0574 

on  principal .9426 .9426 

^9709 

Last  instalment 1 .00 

Interest .0291 

Principal  in  full 9709  .9709 


52 


Thk  Doctrin  of  Interest. 


Thus  the  annuitant  has  received  interest  in  full  on  the 
principal  outstanding,  and  has  also  received  the  entire  original 
principal.  The  correctness  of  the  basis  on  which  we  have  been 
working  is  corroborated. 

116. — It  is  usual  to  form  a  schedule  showing  the  com- 
ponents of  each  instalment  in  tabular  form. 


Total 
Instalment 


Interest 
Payments 


Payments 

on 
Principal 


Principal 
Outstanding 


1904  Mar.  9 

1904  Sept.  9 

1905  Mar.  9 

1905  Sept.  9 

1906  Mar.  1 


1.00 
1.00 
1.00 
1.00 


.1115 
.0849 
.0574 
.0291 


.8885 
.9151 
.9426 
.9709 


3.7171 
2.8286 
1.9135 
0.9709 
0.0000 


4.00 


2829 


3.7171 


•  117. — The  payments  on  principal  are  known  as  amorti- 
zation, which  may  be  defined  as  the  gradual  repayment  of  a 
principal  sum  thru  the  operation  of  compound  interest.  It 
differs  from  the  ordinary  compound  interest  in  this,  that  the 
new  principal  for  each  period  is  less  than  the  previous  one. 

118. — As  an  example  of  logarithmic  evaluation  of  an 
annuity,  take  an  annuity  of  $1,  as  before,  for  50  periods  at  the 
rate  of  .03  per  period.  At  the  beginning  of  the  first  period, 
what  is  its  present  worth,  or  what  should  be  paid  in  one  sum 
for  such  annuity  ? 

i=  .03  r=1.0S        nl        .012  837  224  705  172  (to  15  places) 

60Iy=  .641861235  258,6 
As  we  are  discounting,  not 
accumulating,  we  must  take  _ 

the  cologarithm  —  50  L  1 .  358  138  764  741,4 

and  find  the  number.  Factors  A22  B36  C82  DOS  El 2  F73  G23 

p  =  1.03-5  0=  .228  107  079  790 
D  =  l  — p  =  .771892  920  210 
D-f-.03         =25.729  764  007  0    =P 

This  may  be  proved  down  to  maturity  by  amortization, 
the  schedule  beginning  thus: 


Instalment 

Payments  of 

Payments  on 

Principal 

Interest  at  3% 

Principal 

Outstanding 

25.729  764 

1 

1.00 

.771893 

.228107 

25.501  657 

2 

1.00 

.765  050 

.234  950 

25.266  707 

3 

1.00 

.757  901 

.242  099 

25.024  608 

etc. 

etc. 

etc. 

etc. 

49 

1.00 

.057  404 

.942  596 

.970  874 

50 

1.00 

.029  126 

.970  874 

.000  000 

Special  Forms  of  Annuity.  53 

119. — It  may  be  notist  that  each  payment  on  principal,  or 
amortization  for  one  period,  is  the  present  worth  of  the  instal- 
ment at  the  beginning  of  its  period.  From  this  the  instalment 
of  amortization  may  be  calculated  at  any  point  independently 
of  any  other  figures.  Thus  the  payment  on  principal  in  the 
21st  instalment  of  $1  is  the  present  worth  of  $1.00  in  30 
periods,  or  .  411987  ;  because  at  the  beginning  of  the  21st 
period  there  were  30  instalments  yet  to  come. 

120. — It  will  also  be  notist  that  each  amortization  multi- 
plied by  1 .  03  becomes  the  next  following,  these  being  a  series 
of  present  worths  ;  and  that  thus  they  may  be  derived  from 
one  another,  upwards  or  downwards. 


Speciai.  Forms  of  Annuity. 

121. — The  annuities  heretofore  spoken  of  are  payable  at 
the  end  of  each  period,  and  are  the  kind  most  frequently 
occurring.  To  distinguish  them  from  other  varieties  they  are 
spoken  of  as  ordinary  or  immediate  annuities. 

122. — When  the  instalment  (or  rent)  of  the  annuity  is 
payable  at  the  beginning  of  the  period,  it  is  called  an  annuity 
due,  altho  ''prepaid"  would  seem  more  natural.  It  is  evi- 
dent that  this  is  merely  a  question  of  dating.  The  instalments 
compared  with  those  in  Art.  103  are  as  follows: 


/ 

Immediate 
Annuity 
4  Periods 

Annuity 

Due 
4  Periods 

Immediate 
Annuity 
5  Periods 

1 

1.00 

1.03 

1.00 

1.03 

1.0609 

1.03 

Amounts  of  / 

1.0609 

1.0927 

1.0609 

( 

1.0927 

1.1255 

1.0927 
1.1255 
5.3091 
—1 

4.1836  4.3091  4.3091 

To  find  the  amount  of  an  annuity  due,  for  t  periods, 
find  the  amount  of  an  immediate  annuity  for  t  ■\-\  periods  and 
subtract  $1. 


54  I'hk  Doctrin  of  Interest. 

123. — In  finding  the  present  worth: 


Immediate 
Annuity 
4  Periods 

Annuity 

Due 
4  Periods 

Immediate 
Annuity 
3  Periods 

.888487 
.915142 
.942596 

.915142 
.942596 
.970874 

.915142 
.942596 
.970874 

.970874 

1.00 

2.828612 
+1. 

3.828612         3.828612 

To  find  the  present  "worth  of  an  annuity  due  for  t 
periods  find  the  present  worth  of  an  immediate  annuity 
for  f  —  1  periods  and  add  $1. 

124. — A  deferd  annuity  is  one  which  does  not  commence 
to  run  immediately,  but  after  a  certain  number  of  periods,  as 
an  annuity  of  5  terms,  4  terms  deferd,  which  would  begin  at 
the  fourth  period  from  now  and  continue  to  the  ninth  inclusiv. 

Its  present  worth  is  r^  -\-  r-^  -{-  r^  -\-  r^  -}-  r^ 
An  annuity  of  the  entire   nine   terms  would  be  worth  now 

1  +  ;-!  +  r-2  +  H  +  r-*  +  r^  +  r^  +  r^  +  r^ 
If  from  this  the  value  of  the  four  deferd  terms  be  subtracted 
it  will  leave  the  value  of  the  deferd  annuity. 

125. — To  find  the  present  worth  of  an  annuity  for  m 
terms,  deferd  n  terms,  subtract  from  the  value  of  w  +  « 
terms  that  for  n. 

126. — A  perpetual  annuity,  or  a  perpetuity,  is  one  which 
never  terminates.  Its  amount  is  infinity,  but  its  present  worth 
can  be  calculated  at  any  certain  rate  of  interest.  If  the  rent 
of  the  annuity  is  $1  and  the  rate  is  .  05,  the  value  of  the  annuity 
is  such  a  sum  as  will  produce  $1  at  that  rate  or  $200,  being 
$1  /  .05.  The  compound  discount  is  the  entire  $1,  being  for 
an  infinit  number  of  terms ;  therefore  the  rule  still  holds : 
divide  the  compound  discount  by  the  rate  of  interest. 

127. — Annuities  at  two  successiv  rates  may  occur  ;  say  5 
per  cent,  for  10  years  and  then  4  per  cent,  for  10  more.  The 
second  part  is  evidently  a  deferd  annuity,  and  therefore  its 
present  worth  is  the  same  as 

20  years        at        4% 

less  10  years        at        4% 

+    10  years        at        5% 


The  Unit  of  Time.  55 

128.— In  all  these  examples  of  annuities  it  has  been 
assumed  that  the  term  or  interval  between  payments  is  the 
same  length  of  time  as  the  interest-period.  For  example,  the 
rate  of  interest  may  be  so  much  per  year,  while  the  payments 
are  half-yearly  or  quarterly  ;  or  there  may  be  yearly  payments 
while  the  desired  interest-rate  is  to  be  on  a  half-yearly  basis. 
We  shall  defer  the  treatment  of  these  cases  until  the  subject 
of  nominal  and  effectiv  rates  has  been  discust. 

129. — There  may  also  be  varying  annuities,  where  the 
instalment  changes  by  some  uniform  law.  These  seldom  occur 
in  practice.  Where  the  change  is  simple,  as  in  arithmetical 
progression,  the  annuity  may  be  regarded  as  the  sum  of  several 
annuities,  otherwise  the  values  must  be  separately  calculated 
for  each  term.  An  annuity  running  for  5  terms,  as  follows  : 
13,  18,  23,  28,  33,  may  be  regarded  as  (1)  an  annuity  of  13  of 
5  terms  ;  (2)  an  annuity  of  5,  4  terms ;  (3)  an  annuity  of  5, 
3  terms ;  (4)  an  annuity  of  5,  2  terms ;  (5)  a  single  amount 
of  5.  

The  Unit  of  Time. 

130. — It  makes  no  difference  in  the  result  whether  each 
term  is  a  year,  or  a  month,  or  a  day,  so  long  as  the  number  of 
terms  (/)  and  the  rate  per  term  {t)  are  ascertaind.  But  unfor- 
tunately the  habit  has  been  fixt  in  common  speech  of  stating 
the  rate,  not  at  so  much  per  term,  but  so  much  per  annum, 
even  when  the  interest  is  payable  or  chargeable  semi-annually 
(which  is  the  prevalent  custom),  or  quarterly,  or  monthly, 

131. — When  we  refer  hereafter  to  a  nominal  rate  per 
annum,  we  shall  write  "per  cent."  in  full,  using  for  actual 
rates  per  period  the  symbol  %  or  the  decimal.  The  letters 
a,  s,  q,  or  m,  will  stand  for  "payable  annually,"  "semi- 
annually," "quarterly,"  or  ** monthly." 

132. — The  following  phrases  need  interpretation  into 
more  exact  language: 

(a)  **  Six  per  cent,  per  annum,  payable  annually,"  means 
what  it  says  :  six  per  cent,  per  term,  the  term  being  a  year. 

{F)  "Six  per  cent,  per  annum,  payable  semi-annually," 
means  three  per  cent,  each  half  year  ;  which  is  more  than  six 
per  cent,  per  year. 


56  Tun  DocTRiN  OF  Interest. 

(c)  "Six  per  cent,  per  annum,  payable  quarterly,''  means 
one-and-one-half  per  cent,  per  term  of  three  months. 

(d)  **  Six  per  cent,  per  annum,  payable  monthly,"  means 
one-half  per  cent,  per  month. 

133.— In  cases  (^),  (c)  and  (d),  the ''6"  is  fictitious. 
The  ratios  which  must  be  used  are  1.03,  1.015  and  1.005,  not 
1.06  at  all.  *'  Six  per  cent."  is  known  as  the  nominal  rate, 
but  the  effectiv  rate  for  the  entire  year  is  different. 

Taking  up  the  above  four  cases: 

(a)  Here  the  nominal  and  the  effectiv  rate  are  identi- 
cal;  .06. 

(3)  Here  the  effectiv  rate  is  .03  per  half  year;  for  the 
year  .0609. 

(c)  Here  the  effectiv  rate  is  .  015  per  quarter ;  for  the 
year  .06136355. 

(d)  Here  the  effectiv  rate  is  .005  per  month;  for  the 
year  .06167781. 

134. — Thus  the  words  **  six  per  cent,  per  annum  "  have 
four  different  meanings,  according  to  the  qualifying  phrase 
used,  or  understood.  Let  /  represent  the  nominal  rate  "per 
annum,"  i  being  the  rate  per  term,  and  k  the  effectiv  rate 
per  year. 

Then  in  (a),  where  r  =  1.06  and  /  =  1, 

1  +  y^  =  1  -l-y=  1.06 
In  (5),  where  r  =  1.03,  and  /  =  2, 

1  +  ^  =  ^-^^  (1  +  y2jy=  (1.03)^=  1.0609 

In  (c) ,  where  r  =  1 .  015  and  t  =  4, 

l-{.k  =  r'=(l  +  }ijy=  (1.015)*=  1.06136355  || 
In  (flO,  where  ?-  =  1 .  005  and  if  ==  12, 

l-\-k  =  r^'=  (1  +  i^y)^^=  1.005)^^=  1.06167781 1| 
These  values  may  be  ascertained  by  logarithms  or  by 
arithmetic. 

135. — Case  (3)  furnishes  an  arithmetical  solution  which  is 
very  convenient.  Expanding  (1  •\-j/^y  by  the  binomial  theorem 
we  have  1  +/  +/ V4.  To  the  nominal  rate  the  quarter  of  its 
square  is  to  be  added  to  give  the  effectiv  rate  if  compounded 


The  Unit  of  Tims.  57 

at  half  periods.  Thus  at  6%  for 7,  .06^  =  .0036,  .0036/4  = 
.0009;  .06  +  .0009  =  .0609.  At  8%,  .08^  =  .0064; 
.0064/4=1  .0016.    k=  .0816. 

136. — The  rate  k  being  =/  -{-j^/4,  we  may  factor  this, 
making  ity(l  +y/4).  1  +y/4  is  thus  a  multiplier,  reducing 
the  nominal  rate  payable  semi-annually  to  an  effectiv  annual 
rate.  For  six  per  cent,  this  multiplier  would  be  1.015, 
(. 0609  / .  06) ;  for  five,  1 .  0125  ;  for  four,  1 .  01 ;  for  3%,  1 .  0075  ; 
for  2%,  1.005.  The  same  reasoning  applies  to  a  nominal  half- 
yearly  rate,  payable  quarterly.  If  3%  is  i  for  the  half-year, 
3  (1.0075)  is  y  for  the  half  year  with  quarterly  payments, 
or  3.0225. 

137. — But  the  annual  rate  given  may  be  the  effectiv  rate 
(z)  and  the  question  be,  what  rate  (y)  will  be  equivalent  for 
the  case  of  more  frequent  payments,  giving  k  as  the  nominal 
rate  per  annum  for  that  frequency. 

Case  {a)  is  the  same  as  before. 

Case  (3)  1  +y  =:  (1  +  0^=  (1  +  y2k)  For  i  =  6%, 
l-}-J=:(1.06y/-=l. 02956301  ;  and  /^  =  2j  =  . 05912602. 
That  is^  to  produce  6%  payable  annually,  we  must  invest  at 
5.912602%  per  annum,  payable  semi-annually,  or  2.956301% 
per  period  of  six  months. 

(c)  1  +  y  ==  (1  +  t)H  =a  +  H^)  For  /  =  6%, 
l^  k  =  1.05869538,  payable  quarterly. 

(d)  1  +y  =:  (1  +  i)^  =  (1  +  iW  For  i  =  6%; 
k  =  .058269,  payable  monthly. 

138. —  In  annuity  calculations  the  period  or  interval 
between  cash  payments  is  to  be  considerd  as  well  as  the  fre- 
quency of  compounding  the  interest.  Here,  also,  the  terms 
are  reduced  to  the  **per  annum"  standard.  An  annuity  of 
$50  per  half  year  is  usually  spoken  of  as  an  annuity  of  $100, 
payable  semi-annually.  What  the  actual  value  of  the  yearly 
revenue  is,  depends  upon  the  rate  of  interest  assumed  in  the 
problem. 

139. — If  ^a  represents  the  instalment  or  "rent"  of  the 
annuity  for  each  half-year,  and  i  the  rate  of  interest  for  the 
half-year,  the  equivalent  of  these  two  cash  payments  for  the 


58  Ths  Doctrin  of  Interest. 

year  will  be  Yza  ■}-  }4a  (1  -^  i)  =  a  +  }4ai  =  a  (1  -f  K^')- 
If  y  is  the  nominal  rate  per  annum  or  2i,  then  the  annual 
effectiv  payment  is  a  (1  -\-  j/^)  and  1  +y/4isa  multiplier  for 
transforming  a  yearly  annuity  into  a  half-yearly  one.  This  is 
the  same  multiplier  which  was  alredy  found  to  transform  a 
yearly  nominal  rate  of  interest,  compounded  semi-annually 
into  its  corresponding  ejffectiv  rate.  This  multiplier,  1  +y/4, 
will  be  found  important  in  practis.  It  may  be  called  the 
co-efl5cient  of  double  frequency,  or  C^^^  The  ^^^  represents  the 
ratio  of  the  frequency  of  compounding  to  that  of  payment. 

140. — If  the  rate  of  interest  is  3%"  per  half  year  (6  per 
cent. ,  s)  and  the  annuity  payment  $1  per  annum,  to  find  the 
amount  of  the  annuity  for  four  years,  we  may  reduce  the 
interest  to  the  annual  standard,  the  cash  being  alredy  there. 

The  annual  equivalent  of  the  rate  is  .0609  (6  x  1.015). 
Twice  the  logarithm  of  1 .  03  .  012837224705 

is  log.  1.0609  =  .025674449410. 
The  first  step  is  to  find  the  amount,  for  which  purpose  the 
logarithm  is  multipHed  by  4,  .  1026977976400. 
This  is  also  8  times  the  logarithm  of  1.03,  so  that  we  gained 
nothing  by  squaring  1.03.  From  either  view  the  amount  is 
1 .26677008  and  the  compound  interest  is  .26677008.  This  is 
next  to  be  divided  by  the  rate  of  interest,  which  is  not  .  03, 
nor  .06,  but  .0609. 

.0609). 26677008(4.  3804601,  amount  of  annuity. 
2436 


2317 
1827 

4900 

4872 


2808 
2436 

372 
365 


The  Unit  of  Time.  59 

141. — We  may  test  this  result  as  follows: 

End  of  first  year  ;  cash 1.00000 

third  half  year  ;  interest  .03  on  1.00. .         .03 
♦•        second  year;  "         .03  on  1.03..         .0309 

"  ••  "  cash 1  .00000 

Total 2  .  0609 

End  of  fifth  half  year  ;  interest  .03  on  2.061 . .         .06183 

'•        third  year;         interest  .03  on  2.123. .         .06369 

cash _1^ 

Total 3.18642 

End  of  seventh  half  year  ;  interest .  03  on  3 . 1 86        .09558 

••        fourth  year ;  interest  .03  on  3.282        .09846 

cash J_^ 

Total 4.38046 

142. — We  may  simplify  this  method  a  little  further.  Had 
we  made  the  instalment  60  cents  each  half  year,  the  compound 
interest  would  have  been  half  as  much,  or  .  13338504.  This 
would  have  been  divided  by  .03,  giving  4.446168.  It  would 
have  been  the  same  had  we  divided  the  compound  interest  of 
$1  by  .06.  But  we  did  divide  it  by  .0609,  which  is 
.06  X  1.015,  the  latter  being  the  coeflficient  of  double  fre- 
quency. We  might,  therefore,  have  divided  the  amount  of  the 
annuity  when  payable  semi-annually  by  the  C^^^ 
4.446168/1.015  =  4.38046 

143. — Therefore,  an  annuity  payable  annually  is  trans- 
formd  as  to  its  amount  into  one  payable  half-yearly  by  multi- 
plying it  by  the  C^^). 

144. — The  present  worth  of  the  annuity  is  subject  to  the 
same  law  ;  when  the  annual  payment  is  divided  into  two  equal 
sums  its  present  worth  is  increast  in  the  ratio  of  1  +y/4  or 
1  -j-  z/2.     In  the  case  given  above 

the  logarithm  .  1026977976400 

would  have  been  changed 

to  its  cologarithm  1  .  8973022023600 

the  number  of  which  would 

be  the  present  worth         .  789409234 

The  compound  Discount  would  be    .210590766 
and  the  rate  or  divisor  as  before    .0609 
giving  the  present  worth  of  the 

annuity  as 3  .  45797645 


6o  The  Doctrin  op  Interest. 

145. — The  correctness  of  this  may  be  demonstrated  as 
follows : 

Amount  invested  in  annuity 3.45797645 

Half-year's  interest  on     3.457976  +       .10373929 

3.56171574 
Half-year's  interest  on    3.561716—       .  10685147 

3.66856721 
Annual  instalment 1.00000000 

2  .  66856721 
Half -year's  interest  on     2.668567+       .  08005702 

2  .  74862423 
Half-year's  interest  on     2.748624+       .08245873 

2.83108296 
Annual  instalment 1..  00000000 

1.83108296 
Half-year's  interest  on     1.831083—       .05493249 

1.88601545 
Half-year's  interest  on     1.88  6  015+       .  0  5  658046 

•  1  .  94259591 
Annual  instalment i 1.00000000 

.94259591 
Half-year's  interest  on         .942596—       .02827788  , 

.  97087379 
Half-year's  interest  on     '.,9  708738+       .02912621 

1  .  00000000 
Ivast  instalment 1.0000000  0 

146. — Had  the  payments  been  half-yearly,  each  being  60 
cents,  the  compound  discount  would 

have  been 105295383 

and  we  should  have  divided  by  .03, 

giving 3.5098461 

Dividing  by  the  C^^)  i  oi5,  we  should 

again  have  the  value 3.457976  + 

147. — The  conclusion  is  that  there  are  two  ways  of  calcu- 
lating the  amount  or  present  worth  of  an  annuity  where  the 
interest  compounds  with  twice  the  frequency  of  the  cash 
payments. 

(1)  Procede  as  if  both  were  at  the  greater  interval,  tak- 
ing care  to  use  the  effectiv  rate  of  interest  in  dividing. 

(2)  Procede  as  if  both  were  at  the  smaller  interval,  the 
instalment  being  half  as  much  and  divide  the  result  by  the  C^^). 


The  Unit  of  Time.  6i 

148. — Where  the  interest-period  is  greater  than  the  pay- 
ment-period, or  the  payments  are  made  twice  as  frequently  as 
the  interest  is  compounded,  the  solution  is  less  easy  because  it 
depends  on  evolution. 

149.  — Half  of  the  instalment  is  paid  when  only  half  the 
interest-period  has  elapst.  It  may  be  considered  as  earning 
interest  for  the  other  half -period,  but  the  rate  must  be  taken 
effectivly.  Thus,  if  the  interest  for  a  period  is  .03,  the  ratio 
for  the  half-period  is  the  square  root  of  1.03,  or  1.01488916. 

The  half-instalment  paid  at  the  half- period  becomes  at  the 

end  of  the  full  period 50  X  1 .01488916  =    .50744458 

The  other  half-instalment  is  only .50 

and  the  total  of  both  is 1.00744458 

This  is  the  effectiv  instalment,  insted  of  the  nominal  instal- 
ment, $1.  It  is  a  coejB&cient  of  frequency  as  to  payments,  and 
may  be  represented  by  C^^),  meaning  that  the  interest  is  com- 
pounded only  half  as  often  as  a  payment  is  made. 

150. — If  the  period  of  compounding  is  the  half  year  at  .03 
per  period  (a  nominal  rate  per  cent,  of  .06),  the  effectiv  rate 
per  quarter  is  1.01488916  and  the  G^^)  is  1.00744458,  being 
half  the  square  root  of  1  +  half  the  nominal  rate.  If  the 
nominal  rate  is  3 . 8  per  cent,  (s),  take  first  the  logarithm  of 

1.019 008 174  184  006,4 

and  divide  by  2... 004  087  092  003,2 

The  number  corresponding  to  this 

is ^ 1.009  455  299 

halving  the  decimal  part .   1 .  004  727  649 

is  the  C^^^  for  a  rate  of  3 .8  per  annum  (.y),  payments  quarterly, 
or$l  (^). 

151. — The  coefficient  of  frequency  (C^^O  has  to  be  com- 
puted at  the  commencement  of  each  problem  by  the  above 
method. 

152. — The  computation  of  the  amount,  or  of  the  present 
worth,  as  the  case  may  be,  then  goes  on  just  as  if  the  payments 
took  place  at  the  same  times  as  the  compoundings.  When 
completed,  the  result  is  multiplied  by  O^K 

153. — When  the  interest-period  is  semi-annual  and  the 
instalments  are  paid  quarterly,  it  is  better  to  ignore  the  '  *  per 


62  The  Doctrin  of  Interest. 

annum"  rate  and  treat  of  the  periods  (half  years)  and  half 
periods  (quarters) ,  after  the  commencement, 

154.— An  annuity  of  $2  per  annum,  payable  quarterly, 
interest  to  be  compounded  semi-annually,  for  2  years  at  3i^ 
per  cent,  per  annum,  would  be  stated  as  an  annuity  of  $1  per 
period,  payable  by  half-periods  interest  at  lA^  per  period,  and 
continuing  for  four  periods.  The  present  worth  of  this  annuity, 
omitting  the  condition  ''payable  by  half -periods,"  would  be 
3.81698703,  which  x  1.00472765  is  3.8350324,  the  present 
worth  when  the  annuity  is  paid  at  the  quarters  or  half-periods. 
Tested  as  follows: 

Present  worth 3  .  8350324 

Interest  at  .019 .0728656 

3.9078980 
First  and  Second  Instalments,  with  interest 

on  the  first 1.0047276  -|- 

2  .  9031704 
Interest  at  .019 .055  1602 

2.9583306 
Third  and  Fourth  Instalments,  as  before. . .  1  .0047277 

1.9536029 
Interest  at  .019 .0371185 

1  .9907214 

Fifth  and  Sixth  Instalments 1.0017276 

.9859938 
Interest  at  .019 .0187339 

1  .  0047277 
Seventh  and  Eighth  Instalments 3  .0047277 

The  C<^)  being  almost  exactly  1.00472765,  it  is  taken 
alternately  as  1.0047276  and  1.0047277. 

155. — Values  of  C^^)  for  all  ordinary  rates  are  found  by 
taking  half  the  decimal  part  of  the  figures  under  ' '  Square 
Root "  in  Table  VI.  of  the  Text  Book  of  the  Accountancy  of 
Investment,  Part  III,  the  **  1"  remaining  where  it  is. 

156. — To  find  the  amount  or  the  present  worth  of  an 
annuity  where  half  of  each  instalment  is  collected  midway  of 
the  period,  procede  as  if  the  entire  instalment  were  collected 
at   the  end   and   then  multiply  the    result   by   C^^\    being 

1  +  ^  (^rr^'-i). 


Fractionai.  Periods.  63 

157. — In  some  theoretical  computations  interest  is  con- 
ceived as  compounding  momently  or  continuously.  Interest 
at  6%  per  annum,  when  compounded  momently,  gives  an  equiv- 
alent effectiv  rate  of  .061837.  This  is  obtained  by  multiply- 
ing the  rate  .06  by  the  constant  quantity  .4342944819  (or  as 
many  figures  as  required);  considering  this  as  logarithm, 
its  number  will  be  the  ratio  sought,  1.061836546539.  If  .06 
is  the  effectiv  rate  and  it  is  desired  to  find  the  nominal  rate, 
multiply  the  logarithm  of  the  ratio  (L)  by  the  constant  quan- 
tity 2 .  302585092994,  or  so  much  as  required  and  the  result  will 
be  the  nominal  rate. 

log.  1.06  =  .0253058652648.  This  X  2.30585092994 
=  .0583689  +  .  These  constants  depend  on  the  Naperian 
logarithms. 

Fractionai.  Periods. 

158. — We  have  hitherto  treated  only  of  entire  periods,  but 
it  is  quite  usual  that  the  number  of  periods  should  be  a  mixt 
number,  sometimes  a  fraction  only. 

159. — A  det  is  due  in  one  year  from  now,  at  six  per  cent, 
annually ;  but  the  dettor  has  the  privilege  of  paying  at  the 
half  year  ;  what  interest  should  he  then  pay  ?  There  are  two 
answers  to  this  question,  depending  on  whether  it  is  to  be  con- 
siderd  legally  or  equitably  —  by  simple  interest  or  by  compound 
interest. 

160. — Legally,  the  rate  is  .03  per  half  year,  the  law  not 
recognizing  the  justice  of  compound  interest.  Equitably,  that 
is  not  the  true  proportion  in  which  the  interest  should  be 
divided.  The  creditor  gets,  not  six  per  cent,  annually,  but 
six  per  cent,  semi-annually,  which  we  have  seen  to  be  more 
profitable. 

161. — The  compound  interest  for  a  half  term  is  at  the  rate 
of  .02956301  only,  not  .03.  Compound  interest  for  several 
periods  is  greater  than  simple  interest ;  conversely,  for  part  of 
a  period  the  compound  interest  is  the  lesser. 

162.— If  the  det  spoken  of  is  $1,000,000  and  is  discharged 
at  midyear  by  a  payment  of  $1,030,000,  the  creditor  has  the 


64  The  Doctrin  of  Interest. 

use  for  six  months  of  $30,000,  at  some  rate  from  which  the 
dettor  has  no  benefit,  besides  the  use  of  the  $1,000,000  to 
which  he  is  entitled. 

163. — If  interest  were  not  a  constant  force,  but  a  periodi- 
cal incident,  there  would  be  no  such  thing  as  interest  between 
the  periodical  dates  ;  one  would  have  to  pay  a  full  period  or 
nothing. 

164. — The  result  of  this  inconsistency  is  that,  conven- 
tionally, when  interest  is  calculated  on  a  certain  number  of 
terms  and  a  fraction  of  a  term,  the  interest  compounds  for  the 
integral  terms,  but  remains  simple  during  the  fraction  of  a  term. 

165. — For  four-and-a-half  years  on  the  conventional  inter- 
est plan  at  six  per  cent,  annually,  the  compound  interest  must 

be  calculated  for  four  years  ;  amount 1.26247696 

then  this  must  be  multiplied  by  1.03  (the  con- 
ventional ratio  for  the  half  year)  producing  1.30035127 
This  number  is  exactly  midway, arithmetically, 

between  the  amount  at  four  years 1.26247696 

and  that  at  five  years 1.33822558 

This  plan  of  dividing  the  difference  in  proportion  to  the 
time  elapst  is  generally  used  where  the  even  periodical  values 
can  be  obtaind  from  tables,  especially  in  case  of  valuation  of 
bonds,  as  will  be  shown  hereafter. 

166. — In  scientific  interest,  the  ^  forms  part  of  the  num- 
ber of  terms.    The  log.  1 .  06 025  305  865  264,8 

being  multiplied  by  4.5  gives 113  876  389  191,6 

the  number  for  which  is 1 .29  979  957  070 

This  result  might  have  been  obtaind  by 

multiplying  the  4  year  amount 1 .  26  247  696 

by  the  inconvenient  number 1 .  02  956  301 

which  is  the  effectiv  ratio. 

167. — When  annuities  are  to  be  sumd  or  valued,  it  is 
necessary  to  get  the  value  for  the  entire  terms  first  and  then 
multiply  by  the  effectiv  rate  for  scientific  interest ;  for  conven- 
tional interest  either  multiply  by  the  conventional  rate,  or 
"split  the  difference,"  according  to  time  elapst.  It  is  impos- 
sible to  value  or  sum  the  annuity  in  one  operation  by  a 
fractional  multiplier,  for  the  reason  that  these  processes  depend 
entirely  on  a  uniform  ratio. 


Sinking  Funds.  65 

168. — It  is  the  universal  custom  in  actual  business  to  treat 
parts  of  terms  by  simple  interest,  not  by  compound  ;  conven- 
tionally, not  scientifically. 


Sinking  Funds. 

169. — We  have  hitherto  assumed  the  periodical  instal- 
ment, or  rent  of  an  annuity,  to  be  1.  When  this  is  some  other 
number,  the  amount  or  present  worth  of  $1  is  multiplied  by 
that  other  number ;  that  is,  the  amounts  (or  present  worths) 
are  directly  proportionate  to  the  rent.  But  sometimes  we  have 
given  the  amount  or  the  present  worth  as  a  fixt  sum  and  wish 
to  find  an  instalment  which  will  produce  that  amount  or  ex- 
tinguish that  present  worth. 

170. — We  have  seen  that  the  amount  of  an  annuity  of  $1 
at  3%  for  50  periods  is  $112.79687.  If  the  amount  were 
$1000  insted  of  $112.79687,  it  is  evident  that  each  instalment 
must  be  increast  as  many  times  as  $112 .  79687  is  containd  in 
$1000.  The  quotient  is  8.8655.  Therefore,  under  the  same 
conditions  where  $1  amounts  to  $112.79687,  $8.8655  will 
amount  to  $1000.  If  the  growth  of  the  two  annuities  be  com- 
pared it  will  be  seen  that  at  any  point  the  one  which  is  to 
accumulate  to  $1000  is  8 .  8655  times  as  large  as  the  one  which 
accumulates  to  $112.79687. 

Instalments  Instalments  of 

of  $1  $8.8655 

8  .  8655 

.  2660 

8  .  8655 


1 

.  0000 

.  0300 

1 

2 

.  0300 

.  0609 

1 

3 

.  0909 

.  0927H- 

1 

17.9970 

.  5399 

8.8655 

27  .  4024 

.8220 

8  .  8656 


4  .  1836  37  .  0899 

etc.  etc. 

Therefore,  to  find  the  instalment  which  contributed  each 
period,  will  amount  to  a  given  sum  S,  divide  S  by  the  amount 
of  an  annuity  of  $1. 


66  The  Doctrin  of  Interest. 

171. — Where  an  annuity  is  so  constructed  that  it  shall 
accumulate  to  a  certain  amount  at  a  certain  time,  it  is  called  a 
sinking  fund.  Frequently  the  uniform  periodical  contribu- 
tion is  itself  calld  the  sinking  fund,  and  is  found  in  the  fore- 
going manner. 

172. — Where  the  present  worth  is  the  quantity  given,  the 
process  of  finding  the  uniform  contribution  which  will  gradu- 
ally extinguish  or  amortize  that  present  worth  by  the  aid  of 
compound  interest  is  similarly  performd.  The  fixt  quantity  is 
the  present  worth  of  an  annuity  of  x  dollars  ;  the  given  present 
worth  divided  by  P,  the  present  worth  of  $1  gives  the  instal- 
ment, x^  necessary  to  amortize  it. 

173. — It  is  required  to  find  what  annual  payment  will 
clear  off  $1000  in  50  periods,  allowing  .03  interest.  We  have 
alredy  calculated  that  a  payment  of  $1  per  period  will  pay  off 
$25 .  729764,  with  interest.  $1000  is  38 . 8655  times  $25.729764  ; 
therefore  the  contribution  must  be  $38 .  8655  per  period,  which 
will,  by  forming  a  schedule,  be  found  to  amortize  the  $1000. 
174. — As  a  provision  for  liquidating  indettedness,  or  for 
replacing  vanishing  assets,  sinking  fund  and  amortization  are 
two  different  applications  of  the  same  principle.  Formerly, 
the  terms  were  used  interchangeably,  but  more  recently  they 
are  distinguisht  as  follows: 

175.  — The  sinking  fund  permits  the  det  to  stand 
till  maturity,  but  in  the  meantime  provides  a  fund 
which  at  maturity  pays  off  the  entire  det,  the  Interest 
on  the  original  sum  being  paid  separately. 

176. — The  amortization  plan  accumulates  nothing, 
\iU\.  gradually  reduces  the  det,  applying  to  this  reduc- 
tion all  the  excess  of  the  contribution  over  the  Interest. 
177.  — The  two  operations  which  we  have  performd  show 
that  the  sums  necessary  to  be  set  aside  for  a  det  of  $1000  dur- 
ing 50  periods  at  .  03  are, 

Sinking  Fund $  8.8655 

Amortization 38.8655 

The  difference  is  the $30 

per  period,  which  on  the  sinking  fund  plan  is  required  to  pay 
the  current  interest,  so  that  actually  the  two  methods  of  con- 
tribution come  to  the  same  thing. 


Interest-bearing  Securities.  67 

178. — The  number  of  terms  necessary  for  a  certain  contri- 
bution per  period  to  amount  to  a  certain  principal  may  be 
found,  but  first  the  amount  of  a  single  dollar  must  be  found. 

The  amount  of  the  annuity  is  I/z,  the  total  compound 
interest  divided  by  the  rate  of  interest.  Multiplying  that 
amount  by  the  rate  gives,  therefore,  the  compound  interest. 
Adding  to  this  $1  we  have  the  amount  of  a  single  dollar,  ^  or 
S  X  z  -h  1.     We  then  proceed  as  shown  in  Art.  93. 

Similarly  the  present  worth  of  the  annuity  being  D/t, 
/>  =  1  —  P  X  z,  and  /  may  be  deduced  therefrom. 

179. — The  rate  of  interest  oi  an  annuity  cannot  be  ascer- 
taind  by  any  direct  formula,  as  it  involvs  the  solution  of 
equations  of  higher  degrees. 

180. — A  special  method  for  finding  the  income-rate  of 
securities  by  gradual  approximation  will  be  given  hereafter. 
(Art.  231).  

Interest- BEARING  Securities. 

181. — A  bond  (which  is  the  most  usual  form  of  interest- 
bearing  security)  is  a  complex  promise  to  pay: 

1.  A  certain  sum  of  money  at  a  future  time ;  this  is 
known  as  the  principal,  the  par  or  the  capital. 

2.  Certain  smaller  sums,  proportionate  to  the  principal, 
and  at  various  earlier  times.  These  are  usually  known  as  the 
"interest,"  but  as  they  do  not  necessarily  correspond  to  the 
true  rate  of  interest,  it  will  be  better  to  speak  of  them  as  the 
coupons. 

182. — These  various  sums  are  never  worth  their  face  or 
par  until  the  stipulated  times  arrive,  but  are  always  at  a  dis- 
count. The  principal  is  never  worth  its  face  until  its  maturity; 
the  coupons  are  never  worth  their  face  until  the  maturity  of 
each.  Yet  while  both  principal  and  coupons  are  at  a  discount, 
the  aggregate  may  easily  be  worth  more  than  the  par,  and  it 
is  the  aggregate,  principal  and  coupons  which  is  the  subject  of 
the  valuation. 

183.  —If  the  bond  is  sold  at  par,  the  coupon  and  the  in- 
terest are  equivalent.  Take  a  five  per  cent.  (5)  bond  for 
$10,000,  due  in  5  years,  at  par.     Its  value  consists  of 


68  Thb  Doctrin  of  Interest. 

EXAMPLE  1 

1.  The  present  worth  of  $1000  at  10  periods  at  .025 781 .  1984 

2.  The  present  worth  of  an  annuity  of  |25  per  period, 

10  periods 218.8016 

Aggregate 1000.0000 

EXAMPLE  2 

But  if  the  coupons  were  $30  each,  the  bond  being  **  six  per 

cent,"  the  principal  would  still  be  valued  at : 781 .  1984 

while  the  coupons  would  be  worth 262 .  5619 

Aggregate 1043.7603 

EXAMPLE  3 

If  the  bond  were  a  "four  per  cent."  bond,  the  coupons 

being  $20  each,  the  valuations  would  be,  principal 781 .1984 

coupons 175.0413 

Aggregate 956.2397 

All  the  above  calculations  may  be  made  by  logarithms, 
commencing  with  the  logarithm  (L)  of  1 .  025. 

184. — From  these  computations  we  may  draw  the  follow- 
ing inferences : 

1.  If  the  coupon  rate  is  the  same  as  the  income  rate,  the 
bond  is  at  par. 

2.  If  the  coupon  rate  is  greater  than  the  income  rate, 
the  bond  is  worth  more  than  par. 

3.  If  the  coupon  rate  is  less  than  the  income  rate,  the 
bond  is  worth  less  than  par. 

185. — Rule  I.  Any  bond  may  be  valued  so  as  to  earn 
a  given  interest  rate  by  adding  together 

1.  Present  worth  of  the  principal ; 

2.  Present  worth  of  the  annuity,  consisting  of  all  the 

coupons. 

186. — Representing  the  coupon  rate  or  the  proportion 
which  the  coupon  bears  to  the  principal,  by  c  and  the  value 
of  the  bond  for  $1  by  V 

1  —  /-* 
t 
r^  is  the  only  quantity  which  requires  logarithms  for  its  com- 
putation, which  always  begins  with  L,  the  logarithm  of  r. 
/L  is  the  logarithm  of  r^  and  subtracted  from  zero  is  the  loga- 
rithm of  r^.     In  the  above  example 


Interest-bearing  Securities. 


69 


I,  or  log.  (1  +  0  or  log.  1 .  025  =    .  010  723  865  391,8 
log.  1.025^°  =  /L  =  __.107  238  653  918 
log.  1025-^ «  =  1.892  761  346  082 
1.892  761  346  082 /«       .781198  401727 
Substituting  the  above  value  for  r-^  will  give  the  results  in 
Examples  2  and  3. 

187. — In  the  second  and  third  case  the  correctness  of  the 
figures  may  be  proved  by  forming  a  schedule  of  amortization, 
which,  starting  with  the  present  value,  will  bring  the  value,  up 
or  down,  to  par  at  maturity. 

Six  Per  Cent.  Bond,  Net  Income    025. 


Coupons 

Interest  at  .025 

Amortization 

1043.7603 

30. 

25.0940 

3.9060 

1039.8543 

30. 

25.9964 

4.0036 

1035.8507 

30. 

25.8902 

4.1038 

1031.7469 

30. 

25.7937 

4.2063 

1027.5406 

30. 

25.6885 

4.3115 

1023.2291 

30. 

25.5808 

4.4192 

1018.8099 

30. 

25.4702 

4.5298 

1014.2801 

30. 

25.3570 

4.6430 

1009.6371 

30. 

25.2410 

4.7590 

1004.8781 

30. 

25.1219 

4.8781 

1000.0000 

300. 

356.2397 

43.7603 

Four  Per  Cent.  Bond,  Net  Income  .025. 

Coupons 

Interest  at  .025 

Amortization 

956.2397 

20. 

23.9060 

3.9060 

960.1457 

20. 

24.0036 

4.0036 

964.1493 

20. 

24.1038 

4.1038 

968.2531 

20. 

24  2063 

4.2063 

972.4594 

20. 

24.3115 

4.3115 

976.7709 

20. 

24.4192 

4.4192 

981.1901 

20. 

24.5298 

4.5298 

985.7199 

20. 

24  6430 

4.6430 

990.3629 

20. 

24.7590 

4.7590 

995.1219 

20. 

24.8781 

4.8781 

1000.0000 

200. 

243.7603 

43.7603 

70  Thk  Doctrin  of  Interest 

In  the  six  per  cent,  example  the  amortization  is  subtracted ; 
in  the  four  per  cent,  example  the  amortization  of  discount 
(called  also  accumulation  or  accretion)  is  added.  The  figures 
in  the  two  amortization  colums  are  identical. 

188. — This  is  the  most  natural  method  of  valuation,  and 
for  one  who  only  occasionally  employs  it,  perhaps  the  safest. 
There  are  other  methods  which  in  practis  are  briefer. 


189. — The  excess  over  par  in  the  second  example  (six  per 
cent,  coupons),  $43.7603,  is  known  as  premium. 

In  the  third  example  (four  per  cent,  coupons),  the  value 
is  less  than  par  and  the  difference  is  known  as  discount,  a 
word  which  has  several  meanings.  When  I  have  occasion  to 
speak  of  both  premiums  and  discounts  I  shall  use  the  word 
variance  ;  that  is,  variance  from  par. 

190. — The  difference  between  the  coupon-rate,  or  cash- 
rate,  and  the  interest-rate,  or  income-rate,  is  the  sole  cause  of 
the  variance.  This  difference  will  be  called  the  interest- 
difference. 

.  025  being  assumed  as  the  interest-rate,  and  the  coupon- 
rate  .03  or  .02,  the  interest- difference  is  .005. 

191. — Where  the  coupon  is  .03,  the  .025  may  be  con- 
sidered as  interest  on  $1,  and  each  .005  is  a  future  benefit  or 
extra  profit,  which  should  be  paid  for.  Reduced  to  present 
values,  these  benefits  are  the  present  worth  of  an  annuity  of 
.005  per  period. 

192. — The  present  worth  of  an  annuity  of  .005  for  10 
periods  is  .0437603,  or  on  $1000,  $43.7603,  the  same  variance 
as  found  by  the  previous  process. 

193. — Rule  II.  The  variance  is  the  present  worth  of  an 
annuity  of  the  interest-difference.  When  the  coupon-rate  is  the 
greater,  the  variance  is  added  to  the  par  ;  when  the  coupon- 
rate  is  the  less,  the  variance  is  subtracted  from  par. 

194. — Representing  the  variance  by  Q,  the  second  rule 

may  be  exprest  as  follows: 

1  —  ^' 

or  ^^  (1  —  r-') 


Intkrbst-bearing  Securities.  71 

195. — Multiplying  both  numerator  and  denominator  by 

200  will  not  alter  the  value  of  the  fraction,  hence  it  will  be 

the  same  thing  if  we  use  the  nominal  annual  rates.     Insted 

,    .03—  .025  6  —  5      ,  .  ,   . 

of     ^r^ we  may  use  — = —   which  is  easier. 

In  the  above  example,  No.  2,  the  variance  would  be 
obtaind  thus  : 

Q  =  5-=-^  (1  —  .781198401727 
o 

V  =  1  +  ?-=^  (1—  .78119840172/ 
5 

=  1+  i  (.  218801598273) 
o 

=  14-  .0  43760319655  =  1.0  437603  + 

In  the  third  example,  where  c  =  4 

Q  =  ^^^  (.2188016) 

=  —   i   (.2188016) 
o 

V  =  l—  .0  437603=. 9  562397 
The  results  may  be  carried  to  11  or  12  decimals  if  desired. 

196.— A  third  method  (suggested  by  Mr.  Arthur  S.  Little) 
is  based  upon  the  value  of  a  perpetual  bond.  This,  as  there  is 
no  redemption,  is  merely  a  perpetual  annuity,  or  perpetuity  of 
the  ' '  coupon. ' '  The  value  of  such  a  perpetuity  is  c/i.  A  six  per 
cent,  (s)  bond  to  pay  five  per  cent.  (5)  is  .03/.025  =  6/5  =  1.20, 
and  this  value  is  perpetual,  there  being  no  redemption.  But 
if  it  is  known  that  the  variance  (.20)  will  vanish  10  years  from 
now,  the  value  of  the  bond  is  now  lessend  by  the  present  worth 
of  that  variance. 

197.  —Rule  III.  The  terminant  value  is  the  perpetuity 
value,  minus  the  present  worth  of  the  perpetuity  variance. 

v=  ^_(£_i)(^.) 

In  Example  2, the  perpetuity  value  is  6/5  =  1 .  20 
The  present  worth  of  the  vanishing  quan- 
tity .20  is 1562397 


Remainder 1.0437603 


72  The  Doctrin  of  Interest. 

Here  the  first  step  is  to  obtain  the  perpetuity  value  by 
simple  arithmetic.  The  variance  is,  of  course,  .20.  Then 
r-t==.781 198  401  727  is  found  as  usual  and  multiplied  by  .20, 
giving 156239680+ 

In  Example  3,  the  perpetuity  is 80 

but  the  variance  is  still  .20,  and  its  present 

worth  is .15623968 

which  is  added,  making 95623968 

because  minus  a  minus  is  plus. 

198. — Multiplying  down.  Whichever  of  these  three 
methods  has  been  employd  for  ascertaining  the  value  of  the 
bond  at  a  certain  date,  if  the  successiv  values  for  each  period 
are  expected  to  be  required  (and  they  usually  are) ,  it  is  pre- 
ferable to  find  them  by  schedules  of  amortization  rather  than 
resort  to  independent  logarithmic  calculation  for  each.  A 
thoro  test  of  the  correctness  of  all  the  intermediate  values  is 
the  fact  that  the  series  reduces  to  par  at  maturity.  In  this 
test,  insted  of  a  formal  schedule  in  colums,  all  the  figures  may 
be  brought  into  a  single  colum,  so  that  no  marginal  computa- 
tion may  be  needed.  The  amount  of  each  amortization  is  not 
exprest,  but  implied,  in  the  following  example  of  such  a  single- 
col  um  schedule : 

A  four  per  cent,  (j)  bond  for  4  years    to    net    three    96/100   (5). 
The  r  is  1.0198: 

1.0198  nl  .008  515  007  6315 

X8  _.068  120  061  052  0 

Subtract  from  zero : 1 .931  879  938  95 

(A85  B05  C67  D93  E75  F27) 

Present  worth  of  |1 854  830  361  69 

Compound  discount 145  169  638  31 

Divide  by  .0198 7.331  799  914.75 

Multiply  by  interest  difference 000  2 

Premium .001  466  359  98 

Value 1 .001  466  359  98 

.0198         .01 010  014  663  60 

.009 009  013  197  24 

.0008 .000  801  173  09 

1.021  295  393  91 

Coupon 02 

1.001295  393  91 


MuivTiPLYiNG  Down.  73 

We  save  a  number  of  figures  by  adding  and  subtracting  at  the  same 
time,  putting  a  circle  round  the  coupon  to  indicate  subtraction  : 

Resuming 1 .  001  295  393  91 

.01 10  012  953  94 

.009 9  011658  55 

7)801  036  32 


& 


1.001121042  72 

The  operation  may  be  still  further  abridgd  by  amortizing  the  premium 
only,  but  subtracting  the  interest-difference  only,  not  the  entire  coupon  : 

.001  121041  72(1) 
11  210  43 
^0  089  38 
(2)    896  83 

943  239  36  (2) 
9  432  39 
^^8  489  15 
(2)    754  59 

761  915  49  (3) 
7  619  15 
6  857  24 
(2)    609  53 

577  001  41  (4) 
5  770  01 
5  193  01 
(V)    46160 

388  426  03  (5) 
3  884  26 
^^3  495  83 
(2)   310  74 

196  116  86  (6) 
1  961  17 
1  765  05 
(2)    156  89 

Error  in  decimals 03 

.000  000  000  00(7) 
The  (2)  is  two  places  further  to  the  right  than  in  the  first  procedure. 

199. — Computing  Amortizations.  It  may  sometimes  be 
advisable  to  find  and  verify  at  first  the  instalments  of  amorti- 
zation, leaving  this  series  of  amounts  to  stand,  not  filling  in 
the  remaining  colums  of  the  schedule,  until  required.  The 
obtaining  of  the  amortization  colum  is  remarkably  easy,  as 
shown  in  Art.  119. 


74  The  Doctrin  of  Interest. 

200. — Starting  with  the  premium  or  discount  at  t  periods, 

as  above  explaind,  it  is  next  amortized  to  the  extent  of p 

that  is  the  present  worth  of  the  interest-difference.  This  mul- 
tiplied  by  r  gives  the  next  amortization,  — ^  and  so  on  down. 

201. — In  the  last  example,  the  premium  at  4  years  was 

found  to  be 001  466  359  98 

The  next  amortization  is  simply  the  present  worth  at  8  periods 
of  .  0002.  The  present  worth  of  $1  has  been  found  to  be 
.854  830  36169;  this  x  .0002  =  .000170  966  072  238 
Three  figures  may  be  dropped  from  this  with  safety,  leaving 
the  decimals  as  much  extended  as  in  the  previous  operation. 

1.000 000  170  966  072  (1) 

.01    1709  661 

.009 1538  695 

.0008 136  773 

174  351  210  (2) 

1  743  512 

1  569  161 

139  481 

177  803  355  (3) 

1  778  034 

1  600  230 

142  243 

181  323  862  (4) 
1  813  239 
1  631  915 
145  059 

184  914  075  (5) 
1  849  141 
1  664  227 
147  931 

188  575  374  (6) 
1  885  754 
1  697  178 
150  860 
192  309  166  (7) 
1  923  092 
1  730  782 

153  847 

.  000  196  116  887  (8) 

These    are    the    eight   instalments  of    amortization,  which,  added 
together,  should  equal  the  total  premium. 


Computing  Amortization.  75 

.000  170  966  07 
174  351  20 
177  803  35 
181  323  86 
184  914  07 
188  575  37 
192  309  17 
196  116  89 

.001  466  359  98 

But  if  this  test  were  not  available,  the  last  amortization  would  have 
to  be  multiplied  up  by  1 .0198,  which  should  produce  .000200. 

.000  196  116  887 

1  961  169 

1  765  052 

156  893 


.000  200  000  001 


202. — Small  discrepancies  in  the  last  figure  are  to  be  ex- 
pected and  disregarded ;  therefore,  the  decimals  should  be 
carried  beyond  the  figures  which  are  to  be  utilized. 

203. — Discounting.  A  series  of  values  in  reverse  order, 
beginning  from  maturity,  may  be  obtaind  (without  using 
logarithms)  by  division,  the  interest-ratio  being  the  divisor. 
The  entire  amount  to  be  receivd  on  the  above  bond  is:  princi- 
pal $1,  coupon  .02,  total  $1.02.     This  should  be  divided  by 

1.0198. 

1.02  -V-  1.0198  =  1.000  196  116  89 

which  is  the  value  1  period  before  maturity.     The  coupon  .02 
must  be  added  before  the  second  discounting  process. 

1.020  196  116  89  ^  1.0198  =  1.000  388  426  03 

.02 

1.020  388  426  03 

1.020  388  426  08  -f-  1.0198  =  1.000  597  001  41 

This  is  a  laborious  process,  even  if,  insted  of  dividing,  we  mul- 
tiply, by  the  reciprocal  of  1 . 0198,     . 980584428. 


76 


The  Doctrin  of  Interest. 


1.02 


-|-  coupon 


.980  584  428 
.019  611  689 

1.000  196  117 
.02 

1.020  196  117 

.980  584  428 

.019  611  689 

098  058 

88  253 

5  884 

98 

10 

7 

1.000  388  427 


Table  of  Multiples 


1. 

.980  584  428 

2. 

1.961168  857 

3. 

2.941753  285 

4. 

3.922  337  713 

5. 

4.902  922  142 

6. 

5.883  506  570 

7. 

6.864  090  998 

8. 

7.844  675  427 

9. 

8.825  259  855 

204. — Intermediate  Purchases.  It  happens  very  often 
(perhap?  in  a  majority  of  cases),  that  bonds  are  not  purchast 
on  the  very  day  when  the  interest  is  payable.  In  the  preced- 
ing examples  it  was  supposed  that  exactly  8  or  10  periods 
would  elapse  from  the  purchase  of  the  bond  till  its  maturity  ; 
but  the  purchase  may  have  been  a  month,  or  several  months, 
or  months  and  days,  after  the  beginning  of  the  period. 

205. — We  saw  (Art.  167)  that  an  annuity  cannot  be  valued 
by  the  usual  formula  when  the  number  of  terms  is  a  mixt 
number.  We  must  derive  it  from  the  next  regular  term  or 
interpolate  it  between  the  two  nearest.  It  was  also  explaind 
that  this  interpolation  maybe  done  in  two  ways:  one  by  simple 
proportion,  conventionally,  or  by  compound  interest,  scientifi- 
cally. In  the  present  state  of  knowledge  the  conventional  or 
non-scientific  method  is  establisht  by  usage,  altho  it  works 
injustice  to  the  buyer.  The  difference  is  usually  not  very 
large. 

206. — Let  us  suppose  that  in  the  above  Examples,  in  Art. 
182  the  purchase  had  been  made  9}^  periods  before  maturity, 
that  is,  4  years  9  months. 


Intermediatf  Purchases.  77 

The  value  at  10  periods  (Ex.  2)  is.  . .  1043.7603 

the  value  at  9  periods  is 1039.8543 

the  difference,  or  amortization,  is 3 .  9060 

As  half  the  time  has  elapst,  we  assume 
that  half  the   amortization  has  taken 

effect 1.9530 

and   this  we  subtract  from  the  10- 

period  value 1043.7603 

making 1041.8073 

which  is  exactly  half  way  between  the  9-period  and  the 
10-period  values.  Besides  this,  however,  the  purchaser  must 
pay  one-half  of  the  current  coupon,  or  $15.00,  as  "accrued 
interest,"  the  entire  cost  being  1056.8073.  This  is  called  the 
flat  price,  and  formerly  this  was  the  form  in  which  securities 
were  quoted  at  the  Exchanges  ;  now,  however,  the  quotations 
are  understood  as  so  much  "and  interest,"  meaning  that  the 
accrued  interest  is  made  a  separate  item  in  the  bill. 

207.— Had  the  10-year  value  been  multiplied  by  1.0125 
(half  the  interest  rate)  the  same  flat  value  would  have  been 
obtaind.     1043.7603  X  10125  =  1056.8073 

208-— To  apply  the  scientific  plan,  the  1043.7603  would 
have  been  multiplied  by  1.01242284,  giving  1056.7267 

insted  of 1056.8073 

a  difference  of .0806 

in  favor  of  the  purchaser,  but  he  could  not  claim  it  under  the 
law  and  the  customs  of  the  market. 

209. — As  the  usual  period  is  divided  into  six  months,  and 
as  the  odd  days  are  considerd  as  thirtieths  of  a  month,  the 
amortization  for  each  day  is  1/180  of  that  for  a  half  year.  If 
2  months,  17  days  had  past  after  the  interest-date,  then  1.9530 
must  be  multiplied  by  77/180,  giving  .8359  as  the  proportion- 
ate amortization  for  77  days. 

210. — Thus  the  value  of  a  bond  at  any  date  from  its  issue 
to  its  maturity,  at  some  given  rate  of  interest,  may  be  calcu- 
lated by  first  valuing  it  at  at  two  consecutiv  interest  dates, 
according  to  the  rules  given,  and  then  "splitting  the  differ- 
ence" by  dividing  it  into  180ths. 


78  The  Doctrin  of  Interest. 

211. — Even  if  the  unit  of  time  employed  in  the  coupon- 
payments  and  the  interest-compoundings  be  different,  the  rules 
given  in  the  section  entitled  "  The  Unit  of  Time  "  will  enable 
these  to  be  allowed  for,  if  Rule  I  (Art.  185)  is  used  for  valuing. 

212. —  Intermediate  Balances.  —  When  the  regular 
interest-periods  do  not  coincide  with  the  date  of  the  balance 
sheet,  it  becomes  necessary  to  adjust  the  valuations  for  that 
purpose  in  the  manner  just  described  for  purchases  at  odd 
times. 

213. — If  the  interest-dates  are  May  1  and  November  1, 
and  the  dates  for  balancing  are  January  1  and  July  1,  the  bond 
must,  on  January  1,  have  been  amortized  to  the  extent  of  one- 
third  of  that  from  November  to  May,  conventionally. 

214. — A  six  per  cent,  (s)  bond  for  $1000,  to  yield  five  per 
cent.  (^)  due  Nov.  1,  1925,  is  worth  on 

November  1,  1920 1043.7603 

on  May  1,  1921 1039.8543 

the  amortization   for   6  months  is, 

therefore 3.9060 

For  2  months  it  is  one-third 1 .  3020 

and  this  subtracted  from 1043.7603 

leave  the  value  on  Jan.  1, 1921 1042.4583 

Applying  the  same  method  between 

May  1  and  Nov.  1, 1921 1039.8543 

1035.8507 

3)       4.0036 

1.3345 

1039.8543 

we  have  the  balance,  July  1,  1921 . .     1038.5198 

If  we  take  the  January  value 1042 .  4583 

and  multiply  down  ;   x  .025 26.0615 

1068.5198 

—  Coupon 30. 

we  also  get  the  July  result 1038.5198 

Thus  we  have  the  choice  of  interpolating  each  balance- 
value,  or  having  obtaind  one,  of  multiplying  down  to  maturity, 


Intermediate  Bai^ances. 


79 


which  can  be  done  on  the  conventional,  but  not  on  the  scientific 
plan.     The  resulting  schedule  would  be  as  follows: 


Date 

Collected 

Interest 
at  .025 

Amortization 

Value 

Jan.  1,1921  .... 

Value  at 

5  per  cent. 

basis 

1042.4583 

Julyl,     "    .... 

30.00 

26.0615 

3.9385 

1038.5198 

Jan.  1,  1922.... 

30.00 

25.9630 

4.0370 

1034.4828 

Julyl.     -    .... 

30.00 

25  8621 

4.1379 

1030.3449 

Jan.  1,  1923.... 

30.00 

25.7586 

4.2414 

1026.1035 

Julyl,     "    .... 

30.00 

25.6526 

4.3474 

1021.7561 

Jan.  1,  1924.... 

30.00 

25.5439 

4.4561 

1017.3000 

Julyl,     -    .... 

30.00 

25.4325 

4.5675 

1012.7325 

Jan.  1,  1925.... 

30.00 

25.3183 

4.6817 

1008.0508 

Julyl,     -    .... 

30.00 

25.2012 

4.7988 

1003.2520 

Nov.  1,    "    .... 

20.00 

16.7480 

3.2520 

1000.0000 

The  last  period  is  of  only  4  months,  July  1  to  Nov.  1,  Yz  of 
the  half  year.  The  cash  collected  is,  therefore,  considerd  as 
only  $20,  yz  of  $30  ;  each  previous  coupon  had  included  Yi  of 
the  following  half  year,  and  this  must  now  be  squared  up.  In 
the  colum  "  Interest  at  .025  "  the  procedure  is  peculiar. 

The  16,7480  is  composed  of  two  parts  : 

1.  Yi  of  .025  on  the  $1000  par $16.6667 

2.  .025  infullon  the  $3.2520 .0813 

$16.7480 
215. — To  explain  why  interest  in  full  for  the  half-year  is 
reckond  on  the  premium,  go  back  to  the  normal  schedule  in 
Art.  187,  and  it  will  be  seen  that  the  premium  on  May  1  was 
4.8781.  Now,  on  the  conventional  plan,  based  on  simple 
interest,  this  4 .  8781  should  not  vary  during  the  period  ;  there- 
fore the  interest  ought  to  be  : 

Yz  of  .025   of   4  8781  =  .0813 

which  is  the  same  as  .025  of  ^  of  4.8781  (3.520)  =  .0813 
Hence  in  the  last  or  broken  period  the  variance  from  par 
must  be  treated  as  having  earnd  interest  during  the  entire 
period,  while  the  par  itself  has  only  earnd  interest  for  the 
actual  time,  as  four  months. 

216.— It  will  also  be  notist  that  3.2520  is  ^  of  4.8781,  so 
that  if  we  had  calculated  4.8781  by  discounting,  it  would  have 
been  sufi&cient  confirmation  of  the  preceding  values  to  take  Yz 
of  4.8781  and  compare  it  with  3.2520. 


8o  The  Doctrin  of  Interest. 

217. — It  must  be  rememberd  that  the  periods  introduced 
for  balancing  purposes  are  artificial,  and  that,  strictly  speaking, 
amortization  takes  place  only  at  the  dates  when  interest 
becomes  due.  The  charging  of  part  of  the  coupon,  tho  not 
yet  collected,  is  fictitious,  but  in  each  period  until  the  last,  this 
borrowing  is  compensated  for  by  a  fresh  loan. 

218.— Short  periods,  terminal  or  initial. — It  happens 
sometimes  (altho  it  should  be  avoided)  that  the  bond  does  not 
mature  at  the  end  of  an  interest  period,  but  at  some  previous 
date.  This  gives  rise  to  a  fractional  period,  not  an  artificial 
one  like  those  establisht  for  balancing  purposes  (Art.  212), 
but  an  actual  one,  which  must  be  taken  into  account  in  the 
valuation. 

219. — We  will  take  the  case  of  a  six  per  cent.  (5)  bond  for 
$1000.  issued  Jan.  1,  1921,  and  payable  Nov.  1,  1925,  interest 
payments  January  and  July  1,  and  valued  to  pay  five  per 
cent.  (5).  There  are  9  full  periods  and  a  short  period  of  4 
months,  or  ^  of  a  period.  The  coupon  for  this  short  period 
would  be  $20  insted  of  $30,  as  in  such  cases  the  last  coupon  is 
always  proportional  to  its  time.  The  interest  ratio  is  also 
reduced  for  that  period  to  1  +  ^  of  .025,  or  1.016^  by  the 
conventional  plan. 

220. — Using  the  first  method  of  evaluation,  the  following 
are  the  components  of  the  value  : 

An  annuity  of  9  terms  of  |30  each  at  .  025 $239 .  1260 

The  present  worth  of  $1020  at  .025  for  9  terms 
and  .016^  for  1  term. 

$1  for  9  terms .8007284 

Divided  by  1 .  016% .7876017 

Multiply  by  1020 803.3537  v 

1042.4797 
It  will  be  observd  that  this  differs  slightly 
from   the  value  obtaind  in  Art.  209  for  the 
same  length  of  time,  but  with  interest  May 
and  November 1042 .  4583 

The  $1020  referd  to  is  composed  of  the  principal  and  the 
last  or  partial  coupon. 

221. — To  divide  by  a  number  like  1 .016^,  it  is  easier  first 
to  multiply  both  divisor  and  dividend  by  3,  converting  each 
into  a  whole  number. 

1.0116^3       X3        -=3.05 

.8007284    X3        =2.40218508 
2 .  40218508  --  3 .  05  =    .  78760167 


Short  Periods,  TerminaIv  or  Initiai.  8i 

222. — To  illustrate  the  case  of  an  initial  short  period, 
suppose  that  the  above  bond  had  been  issued  on  October  1, 
1920,  3  months  earlier  than  has  been  assumed  ;  issued  Oct.  1, 
1920,  interest  January  and  July,  principal  maturing  Nov.  1, 
1925.  There  is  then  a  preliminary  coupon  for  3  months,  $15, 
to  be  discounted  at  1 .  0125  ;  9  coupons  of  $30  each  forming  an 
annuity  ;  one  coupon  of  $20  and  the  principal,  $1000,  dis- 
counted for  9  periods  at  1.025,  and  one  period  at  1.016^. 
The  value  on  Jan.  1,  1921,  obtaind  as  before,  is  1042.4797. 
The  simplest  way  will  probably  be  to  add  to  this  the  initial 
coupon  $20,  and  discount  back  the  entire  1062.4797  by  divid- 
ing by  1.015,  giving  1046.7780. 

223. — We  may  have  two  other  complications :  the  bond 
may  be  purchast  within  one  of  the  odd  periods  ;  the  balancing 
period  may  be  at  still  another  date. 

224  — If  the  above  bond  were  bought  on  Dec.  1,  1920,  the 
price  would  be  between  1046.7780,  the  October  value,  and 
1042.4797,  the  January  value,  a  three  months'  interval.  The 
difference  is  4 .  2983  ;  and  either  Yz  of  this  (1.4328)  may  be 
added  to  the  January  value  or  ^  (2.8655)  subtracted  from 
the  initial  value. 

1042.4797  +  1.4328  =  1043.9125 
1046 .  7780  —  2 .  8655  =  1043 .  9125 

225. — The  adjustment  of  values  to  balancing  dates  pre- 
sents no  special  difi&culty,  being  performd  by  simple  proportion. 

226. — Cash  payments  on  principal. — Bonds  at  the  same 
rates  of  coupon  and  of  interest,  tho  at  different  dates  of  matu- 
rity, may  be  combined  into  one  schedule.  This  may  be  done 
even  if  the  interest-dates  are  different,  but  it  is  practically 
better  in  that  case  to  keep  the  schedules  distinct. 

227. — We  must  commence  with  an  aggregate  value,  made 
up  of  the  separate  values  of  the  groups  of  bonds  maturing  on 
the  same  day. 

228. — $2000  six  per  cent.  (5)  bonds,  maturing  as  follows  : 
$1000  on  Nov.  1,  1923,  $1000  on  Nov.  1,  1925  ;  interest  .025, 
interest  payments  May  and  November.  Required,  value  on 
Nov.  1,  1920. 

The  value  of  the  first  bond  is 1018.8099 

The  value  of  the  second 1043.7603 

Aggregate  value 2062.5702 


82 


The  Doctrin  of  Interest. 


This  value,  multiplied  down  in  the  usual  manner,  gives  the 
following  schedule.  At  the  date  of  the  maturity  of  bond  No. 
1,  the  cash  colum  must  contain  not  only  the  coupon  $60,  but 
the  $1000  payable  on  principal. 


Date 

Cash 
Collections 

Interest 
at  .025 

Payments 
on  Principal 

Investment 
Value 

1920  Nov.  1 

2062.5702 

1921  May  1 

60.00 

51.5643 

8.4357 

2054.1345 

"    Nov.  1 

60.00 

51.3534 

8.6466 

2045.4879 

1922  May  1 

60.00 

51.1372 

8.8628 

2036.6251 

••    Nov.  1 

1060.00 

50.9156 

1009.0844 

1027.5407 

1923  May  1 

30.00 

25.6885 

4.3116 

1023.2291 

•♦    Nov.  1 

30.00 

25.5808 

4.4192 

1018.8099 

etc. 

etc. 

etc. 

etc. 

229. — The  remainder  of  the  schedule  continues  as  in 
Article  187. 

230. — For  intermediate  balances  (Article  214),  the  interest 
requires  adjustment  at  the  date  of  partial  payment.  We  now 
assume  that  the  above  group  of  bonds  is  to  be  valued  on 
January  1  and  July  1  of  each  year.  The  values  of  the  bonds 
under  consideration  for  January  1,  1921,  would  be,  by  inter- 
polation       1017.3000 

and 1042.4583 


Afir2're2'ate. 

.      20.^9  7.^ft.^ 

Dates 

Cash 
Collections 

Interest 
at  .025 

Payments 
on  Principal 

Investment 
Value 

1921  Jan.    1 

2059.7583 

-    July   1 

60.00 

51.4940 

8.5060 

2051.2523 

1922  Jan.    1 

60.00 

51.2813 

8.7187 

2042.5336 

'•    July  1 

60.00 

51.0633 

8.9367 

2033.5969 

1923  Jan.    1 

1050.00 

42.5066 

1007.4934 

1026.1035 

-    July   1 

30.00 

25.6526 

4.3474 

1021.7561 

etc. 

etc. 

etc. 

etc. 

^31. — The  entries  under  January  1,  1923,  are  peculiar. 
The  $1000  paid  off  was  only  in  possession  for  4  months,  Yi 
of  a  period;  therefore,  $20  is  the  appropriate  sum  to  be  con- 
siderd  as  paid  with  it,  and  if  it  was  kept  in  a  separate  account, 
that  is  all  which  would  be  allocated  to  it.  The  other  $1000  is 
on  interest  during  the  full  period  and  $30  is  charged  to  it. 


Short  Periods,  Terminai,  or  Initial.  83 

Cash  entries : 

For  bond  No.  1  par 1000 

Gross  interest  thereon,    .03,  4  months. .         20 
Gross  interest  on  No.  2,  .03,  6  months._^ 30 

1050 
Interest  entries : 

Bond  No.  1,  Yz  of  period  at  .025 16.6667 

Bond  No.  2,  full  period,  (Art.  210),  at 

.025  on  1033.5969 25.8399 

42.5066 
Applied  on  principal : 

1050.00-42.5066== 1007.4934 

232. — While  this  procedure  may  be  applied  where  the 
successiv  partial  payments  on  principal  are  of  the  most  irregu- 
lar amounts  and  intervals,  their  chief  utility  is  in  what  are 
known  as  serial  bonds,  where  regular  payments  of  principal 
are  made,  usually  annually.  Each  bond  or  group  of  bonds,  as 
it  is  dropt  from  the  total,  carries  with  it  the  appropriate  cash 
and  interest  entries  exactly  as  exemplified  above. 


To  Find  thb  Income  Rate. 

233. — When  the  cash-rate,  the  time  and  the  price  of  the 
bond  are  known,  it  is  very  desirable  to  know  what  is  the 
income-rate,  for,  of  course,  every  one  wishes  to  get  the  highest 
income,  security  being  equal. 

234. — There  is  no  positiv,  direct  method  of  doing  this 
beyond  three  periods,  as  an  equation  of  a  higher  degree  is  not 
directly  soluble.  There  are  only  methods  of  approximation 
and  trial. 

235. — When  printed  tables  are  accessible  it  is  easy  to 
make  a  rough  approximation  by  observing  between  what  values 
the  given  price  lies.  The  smaller  the  interval  between  the 
rates  of  the  table,  the  closer  is  the  approximation,  and  an  addi- 
tional decimal  may  be  obtaind  by  proportion.  With  an  ex- 
tended table  at  close  intervals,  the  result  is  suflSciently  accu- 
rate for  commercial  purposes. 

236. — There  is  a  method  devised  by  the  author  which  will 
produce  even  greater  accuracy,  up  to  12  places,  at  the  expense 
of  considerable  labor.  It  appears  in  the  later  editions  of  his 
"  Text- Book  of  the  Accountancy  of  Investment,"  from  which 
it  is  here  quoted. 


$4  The  Doctrin  of  Interest. 


To  Find  the  Income  Rate. 

1. — Given  a  bond  on  which  there  is  a  premium  or  discount 
Q,  cash-rate  c  payable  in  n  periods,  what  is  the  income-rate,  /? 

2. — Every  premium  or  discount  is  the  present  worth  of  an 
annuity  of  n  terms,  each  instalment  of  which  is  the  difference 
of  rates  ;  or  it  is  the  difference  of  rates  X  such  an  annuity  of 
$1  (Art.  193).  Writing  P  for  the  present  worth  of  an  annuity 
of  $1  (Art.  108):  Q  =  P  X  {c  —  i).  If  instead  of  the  pre- 
mium on  $1  we  use  that  on  $100,  we  have  100  Q  =  P  X 
(100^  —  100/).  It  will  not  affect  the  value  of  the  right-hand 
side  if  we  halve  one  factor  and  double  the  other.  100  Q  = 
^  P  X  (200^  —  2000.  200^  is  the  rate  per  cent,  per  annum 
as  conventionally  termed.  Thus  we  pay  4  per  cent,  per 
annum,  meaning  .02  per  period.     So  also  of  200/  for  i. 

3. — It  is  evident  that  if  we  divide  lOOQ,  the  premium  on 
$100,  by  %P  (which  we  will  hereafter  call  the  trial-divisor), 
we  shall  find  the  difference  of  rates.  But  as  the  annuity  de- 
pends on  the  unknown  rate,  this  does  not  help  us  at  all. 

4. — Let  us  assume  the  rate  of  income  per  annum  to  be  any 
rate  whatever,  and  calculate  the  trial- divisor  at  that  rate. 
Then  there  is  this  property  :  If  the  assumed  rate  is  too  large, 
the  quotient  or  difference  of  rates  will  be  too  small,  and  yet  will 
be  nearer  the  truth,  and  vice  versa.  From  this  approximate 
difference  of  rates  we  derive  a  new  rate  and  proceed  with  this 
as  a  trial-rate. 

The  result  of  this  trial  will  give  a  new  rate  still  nearer 
and  so  on.  We  may  slightly  modify  any  rate  to  make  it  more 
easy  to  work.  If  we  select  our  first  trial-rate  near  the  true 
rate,  fewer  successiv  approximations  will  be  necessary. 

5. — Fortunately  for  our  purpose,  any  table  of  bond  values 
will  readily  give  the  trial- divisor,  by  taking  the  difference 
between  the  values  at  the  same  income-rate  of  two  successiv 
$100  bonds,  say  a  3%  and  a  4%,  a  5%  and  a  6%,  always  1% 
apart. 

6. — For  example,  a  6%  bond  for  $100  (semi-annual)  for 
50  years  is  sold  at  133 .  00,  what  is  the  income  rate  ? 


To  Find  the  Income  Rate.  85 

7. — With  so  large  a  premium  as  33,  the  income-rate  is 
evidently  much  less  than  6.  Let  us  assume  4.  Then  from 
any  bond  table  we  find  on  the  4%  line  the  value  of  a  5%  bond 

to  be 121.55 

and  that  of  a  4%  bond  to  be 100  00 

The  first  trial-divisor  is  therefore 21 .  55 

33.00 -f- 21. 55  =  1.531.  the  difference  of  rates.  6  —  1.531  = 
4.469,  the  new  trial  rate.  Taking  4.45  as  more  convenient, 
the  new  trial-divisor  is  19.98.  33.00-^-19.98  =-  1.651. 
6  —  1.651  =  4.349.  We  find  that  20.315  is  the  trial-divisor 
for  4.35.  33.00  ->  20. 315  =  1.6244.  6  — 1.6244  =  4.3756. 
Next  using  4.37,  trial-divisor  20.25  :  33.00  -f-  20.25  =  4.37, 
almost  exactly,  so  that  4. 37  has  reproduced  itself.  The  value 
of  the  bond  at  4.37,  as  computed  by  logarithms,  is  133.0069, 
an  error  of  less  than  one  cent. 

8. — It  will  be  noticed  in  the  foregoing  example  that  the 
results  always  swing  to  the  opposit  side  of  the  true  rate  ;  that 
is,  if  the  trial-rate  is  too  large  the  next  rate  is  too  small,  and 
the  true  rate  is  between  them.  The  successiv  rates  were 
4. .  .4.469.  .  .4.349. .  .4.3756.  .  .4.37.  4.37  lies  between  any 
pair  of  these.  This  is  always  the  case  with  bonds  above  par. 
With  bonds  below  par  it  is  different.  The  true  rate  always 
lies  beyond  the  approximation. 

9. — As  an  example  of  a  bond  below,  take  a  3%  bond  pay- 
able in  25  years.  If  purchased  at  88. 25,  what  is  the  income- 
rate  ?  The  following  may  be  the  steps,  the  dividend  being 
always  11.75,  the  discount. 

Trial-rates 3.70  3.725  3.7265 

Trial- Divisor 16 .  2190      16 .  175        16 .  17245 

Result 3.7244        3.7264        3.7265 

As  3 .  7265  reproduces  itself,  it  must  be  correct  to  the  4th 
decimal ;  and  the  value  of  a  3%  bond  for  25  years  to  yield 
3.7265  is  found  by  logarithms  to  be  88.25015. 


86  The  Doctrin  of  Interest. 

237. — Since  the  publication  of  the  above  method  some 
simplifications  have  been  suggested,  referring  particularly  to 
the  employment  of  Sprague's  Extended  Bond  Tables. 
Insted  of  obtaining  a  divisor  as  shown  in  par.  5,  we  may  first 
multiply  Q  itself  by  the  interest-difference  and  then  use  the 
entire  variance  at  the  trial-rate  (Q'')  as  the  divisor,  giving  the 
same  result.  Thus  in  par.  7,  where  the  first  assumed  rate  is 
4%  and  the  interest  difference  2,  finding  from  the  Table  that 

the  premium  is 430.983.52 

33  X  2 -^43.098352  =    1.531 

insted  of         33-^^21.55  =    1.531 

238. — The  valuable  suggestion  has  further  been  made  by 
Mr.  E.  S.  Thomas  that  by  using  the  two  nearest  income  rates 
and  interpolating,  five  decimals  may  be  obtaind  at  once.  In 
the  example  where  Q  =  33.00,  200^  =  6,  select  4.35  as  200i^ 
and  4.40  as  200/2.  From  the  Tables  we  find  opposit  4.35, 
335,201.06  and  opposit  4.40,  322,371.36,  the  interest-differences 
being  1.65  and  1.60. 

4 .  35%      33 .  00  X  1 .  65  -^  33 .  520106  =  1 .  624398 

4 .  40%       33 .  00  X  1 .  60  -^-  32 .  237176  =  1 .  637861 

4 .  35  -f  1 .  624398     =     5 .  974398  =  6 .  —  025602 

4.40  +  1.637861     =     6.037861  =  6. +  .037861 
Difference  in  error  0 .  063463 

239. — If  the  same  operation  be  performd  on  a  third 
rate,  4.45  : 

4.45%  33.00  X  1.55  -h  30.974371  =  1.651333 

4.40%  33.00  X  1.60  -f-  32.237176  =  1.637861 

4.45+1.651333     =     6.101333 

4.40  +  1.637861     =     6037861 


Difference  .063472 

which  differs  so  slightly  from  .063463  that  further  differenc- 
ing may  be  neglected. 


To  Find  the  Income  Rate.  87 

240. — By  proportion,  we  may  now  ascertain  at  what  rate 
the  coupon  will  become  6.  From  4.35  to  4.40,  an  extent  of 
.05,    it  varies  by    .06347. 

Therefore  .06347  :   .025602  :  :   .05  :   .0201686 

Add  4.35 

Rate  required  4.3701686 

or  safely  4.37017 

241. — In  the  other  example  (par.  9),  where  Q  =  11.75, 

it  appears  from  the  Tables,  (p.  10) ,  that  the  nearest  rates  are 

3.70%  and  3.75%;  the  corresponding  values  being  88.646703 

and  87.900397  and  the  discounts  11.353297  and  12.099603. 

3 .  70%        —  11 .  75  X  .  70  -f- 11 .  353297  =  —  .  72445916 

3.75%        —  11  75  X  .75  -^  12.099603  =  —  .72832968 

3 .  80%        —  11 .  75  X  .  80  -f- 12 .  837828  =  —  .  73221109 

Diff. 
3.70  —  .72445916  =  2.97554084 
3.75—  .72832968  =  3.02167032  -04612948 
3 .  80  —  .  73221109  -=  3 .  06778891    04611859 
3  —  2.97554084  -=  .02445916 

.04612  :   .02446916  :  :   .05  :   .02650169 

3.70 


3.72650169 
or  safely  3.7265 

242. — The  process  may  be  still  further  abridged  by  mak- 
ing the  interest-difference  the  standard  of  comparison  insted 
of  the  coupon,  giving  the  same  result.     Thus  in  238,  insted  of 
4.35  +  1.624398  =  5.974398  =  6.  -  .025602 
4.40  +  1.637861  =-6.037861  =  6.  +  -037861 

.063463 
might  have  been  written 

1.65  —  1.624398  =  +  .025602 

1.60  —  1.637861  =  —  .037861 


And  in  241 

.70  —  .72445916  =  —  .02445916 

•75  --  .72832968  =  +  .02167032 

.80  —  -73221109  =   +  .06778891 


.063463 

.04612948 
.04611859 


88 


INTEREST  FORMULAS. 


i  =  Rate  of  Interest,  or  the  Interest  on  Unity  for  1  period. 

r~   1  +  /,  or  the  Ratio  of  Increase. 

/  =^  Number  of  Periods  of  Time. 

r*^  =  (1  +  0*  =  Amount  =  s. 

rt^-=  (i  +  /)-t  =  -^^^^  Present  Worth  = /. 

r^  —  1  =  Compound  Interest  =  I. 

1  —   r-^  =  Compound  Discount  =  D. 

Amount  of  Annuity  —  l-^-r-^-r^-^-r^ +  r^-^  =  l/t  =  S. 

Present  Worth  of  Annuity  =  1  +  r^  +  r^  +  r^  . .  r^-^=  B/i  =  P. 

y  =  Nominal  Rate  Per  Annum.     Coefficient  of  Double  Fre- 
quency =  C(^)  =  1  +  V*-     Coefficient  of  Half  Frequency 

=  c(^^)  =  i  +  y2  (y"rT7— 1) 

Sinking  Fund  =  1/S  =  2*/I. 

Amortization    =  1/  P  =:=^  /*/  D  =  //I  +  t. 

c  =  Coupon  Rate  of  Bond,  or  Cash  Rate. 

V  •  =  Value  of  Bond  at  c  to  Earn  i  —  r'^  -\-  cV. 

or  =  1  +  {c—i)  P. 

or=  f-(f-l)P. 


UE  ON  THE  LAST  DATE 


-^\oP^t 


^^^^i---°^ 


;-lfr/^^.te.e-^h 


^iA:;t?S\*'"^^ 


^^.602^ 


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